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OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/10674 | A | OMC225(A) | 100 | 293 | 324 | [
{
"content": "ã$r = \\lfloor r \\rfloor + \\lbrace r \\rbrace$ ãçšããŠäžåŒãå€åœ¢ãããšïŒ\r\n$$\r\n\\begin{aligned}\r\n\\dfrac{1}{\\lbrace r \\rbrace} + \\dfrac{1}{\\lfloor r \\rfloor} = \\dfrac{25}{4r}\r\n&\\iff\\dfrac{r}{\\lbrace r \\rbrace} + \\dfrac{r}{\\lfloor r \\rfloor} = \\dfrac{25}{4} \\\\\\\\\r\n&\\iff\\dfrac{\\lfloor r \\rfloor + \\lbrace r \\rbrace}{\\lbrace r \\rbrace} + \\dfrac{\\lfloor r \\rfloor + \\lbrace r \\rbrace}{\\lfloor r \\rfloor} = \\dfrac{25}{4} \\\\\\\\\r\n&\\iff \\dfrac{\\lfloor r \\rfloor}{\\lbrace r \\rbrace} + \\dfrac{\\lbrace r \\rbrace}{\\lfloor r \\rfloor} + 2 = \\dfrac{25}{4}\\\\\\\\\r\n&\\iff \\dfrac{\\lfloor r \\rfloor}{\\lbrace r \\rbrace} + \\dfrac{\\lbrace r \\rbrace}{\\lfloor r \\rfloor} = \\dfrac{17}{4}\r\n\\end{aligned}$$ \r\nãåããïŒ$\\dfrac{\\lfloor r \\rfloor}{\\lbrace r \\rbrace} = t$ ãšãããšïŒ$t + \\dfrac{1}{t} = \\dfrac{17}{4}$ ãšãªãïŒããã解ãããšã§ $t = 4, \\dfrac{1}{4}$ ãåŸãïŒ$0 \\lt \\lbrace r \\rbrace \\lt 1$ïŒåã³ $\\lfloor r \\rfloor$ ãæ£æŽæ°ã§ããããšãåãããŠïŒ\r\n$$(\\lbrace r \\rbrace, \\lfloor r \\rfloor) = \\left (\\dfrac{1}{4} \\ , 1 \\right), \\left (\\dfrac{1}{2} \\ , 2 \\right ), \\left (\\dfrac{3}{4} \\ , 3 \\right )$$ \r\nãåããïŒãã£ãŠïŒ$r$ ãšããŠããåŸãå€ã®ç·å㯠$\\dfrac{15}{2}$ ãšãªãïŒç¹ã«è§£çãã¹ãå€ã¯ $\\mathbf{17}$ ãšãªãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/10674"
}
] | ãæŽæ°ã§ãªãïŒ$1$ 以äžã®å®æ° $r$ ã§ãã£ãŠïŒä»¥äžã®çåŒãã¿ãããã®ã®ç·åãæ±ããŠãã ããïŒ
$$\dfrac{1}{\lbrace r \rbrace} + \dfrac{1}{\lfloor r \rfloor} = \dfrac{25}{4r}$$
ããã ãïŒçãã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a + b$ ã®å€ã解çããŠäžããïŒãŸãïŒæ£ã®å®æ° $x$ ã«ã€ã㊠$\lfloor x \rfloor$ 㧠$x$ ã®æŽæ°éšåïŒ$\lbrace x \rbrace$ 㧠$x$ ã®å°æ°éšåãè¡šããã®ãšããŸãïŒ |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/10611 | B | OMC225(B) | 400 | 82 | 129 | [
{
"content": "ãäžè§åœ¢ $ABD, ACE, BEH, CDH$ ã¯ããããçŽè§äºç蟺äžè§åœ¢ã§ããïŒ$BE = a, ~ CD = b$ ãšãããšïŒ\r\n$$ AB = 2a + \\sqrt2b, \\quad AC = \\sqrt2a + 2b $$\r\nã§ããïŒåè§åœ¢ $AEHD$ ã®é¢ç©ã¯\r\n$$ 5 = \\frac12 (a + \\sqrt2b)^2 - \\frac12 b^2 = \\frac12(a^2 + b^2) + \\sqrt2 ab$$\r\nã§ããïŒãŸãïŒäžè§åœ¢ $ABC$ ã®å€å¿ã $P$ïŒèŸº $BC$ ã®äžç¹ã $M$ ãšãããšïŒåè§åœ¢ $BPCO$ ã¯æ£æ¹åœ¢ãšãªãã®ã§ïŒäžç·å®çãã\r\n$$ AO^2 = 2AM^2 + 2 PM^2 - AP^2 $$\r\nãšãªãïŒ$BC = 2m$ ãšãããš $PM = m , ~ AP = \\sqrt2 m$ ããã³\r\n$$ 2AM^2 = AB^2 + AC^2 - 2m^2 = 6(a^2 + b^2) + 8\\sqrt2 ab - 2m^2$$\r\nã§ããïŒäžè§åœ¢ $BEC$ ã§ã®äžå¹³æ¹ã®å®çãã\r\n$$ 4m^2 = 2(a^2 + b^2) + 2\\sqrt2 ab $$\r\nã ããïŒ\r\n$$ 49 = AO^2 = 2AM^2 =5(a^2 + b^2) + 7\\sqrt2 ab $$\r\nãåŸãïŒä»¥äžããïŒ\r\n$$\r\n\\left\\lbrace\r\n\\begin{aligned}\r\n& a^2 + b^2 + 2\\sqrt2 ab = 10 \\\\\\\\\r\n& 5(a^2 + b^2) + 7\\sqrt2 ab = 49\r\n\\end{aligned}\r\n\\right.\r\n$$\r\nã解ãã°ããïŒ\r\n$$a^2+b^2 = \\frac{28}{3}, \\quad \\sqrt2 ab = \\frac{1}{3} $$\r\nãåŸãïŒãããã£ãŠïŒ\r\n$$BC^2 = 2(a^2 + b^2) + 2\\sqrt2 ab = \\frac{58}{3} $$\r\nãªã®ã§ïŒè§£çãã¹ãå€ã¯ $\\mathbf{61}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/10611"
},
{
"content": "ãå
¬åŒè§£èª¬ãšã¯ç°ãªãæåã®äœ¿ãæ¹ã ãïŒ$AC=b, AB=c$ ãšããïŒ\r\n\r\nãå
¬åŒè§£èª¬ãšåæ§ã«ïŒçŽè§äºç蟺äžè§åœ¢ãããŸã掻çšããŠïŒåè§åœ¢ã®é¢ç©ã®æ¡ä»¶ã¯æ¬¡ã®ããã«æžãæãããã\r\n$$\\tag {1} 20=-b^2-c^2+2 \\sqrt{2}bc$$\r\n\r\nã次ã«æ¡ä»¶ $OA=7$ ã掻çšãããïŒ\\\r\nã$\\angle BOC=90^{\\circ}$ ãªã®ã§ïŒç¹ $O$ ãäžå¿ã« $\\triangle ABC$ ã $90^{\\circ}$ å転ãããŠïŒç¹ $B$ ãç¹ $C$ ã«äžèŽããããã«ã§ããïŒãã®ããã«å転移åãããšãã®ç¹ $A$ ã®ç§»åå
ã $A^{\\prime}$ ãšãããïŒãã®ãšã $\\triangle ACA^{\\prime}$ ã«ã€ããŠïŒä»¥äžã®ããšããããïŒ\r\n- å転移åãªã®ã§ $A^{\\prime}C=AB$\r\n- $\\triangle AOA^{\\prime}$ ã¯çŽè§äºç蟺äžè§åœ¢ãªã®ã§ $AA^{\\prime}=7\\sqrt{2}$\r\n- è§åºŠèšç®ã«ãã $\\angle ACA^{\\prime}=135^{\\circ}$\r\n\r\nããã£ãŠäœåŒŠå®çãã $98=b^2+c^2+\\sqrt{2}bc$ ãåŸãïŒ\\\r\nãåŒ $(1)$ ãšé£ç«ãããŠïŒ\r\n$$b^2+c^2=\\dfrac{176}{3}, \\sqrt{2}bc=\\dfrac{118}{3}$$\r\nã$\\triangle ABC$ ã«äœåŒŠå®çãçšããŠïŒ\r\n$$BC^2=b^2+c^2-\\sqrt{2}bc=\\dfrac{176}{3}-\\dfrac{118}{3}=\\dfrac{58}{3}$$\r\n\r\n---\r\n\r\nïŒãããªãå¥è§£ïŒ$AO=7$ ã®æŽ»çšæ¹éã ãèšãïŒ\\\r\nãç¹ $O$ ãäžå¿ã« $\\triangle ABC$ ã $180^{\\circ}$ å転ãããŠïŒç¹ $A,B,C$ ã®ç§»åå
ããããã $A^{\\prime}, B^{\\prime}, C^{\\prime}$ ãšããïŒ$AC$ ãš $A^{\\prime}B^{\\prime}$ ã®äº€ç¹ã $T$ ãšãããšïŒ$\\triangle ATA^{\\prime}$ ã«ã€ããŠïŒä»¥äžã®ããšããããïŒ\r\n- åè§åœ¢ $BCB^{\\prime}C^{\\prime}$ ã¯æ£æ¹åœ¢ïŒç¹ã« $BC=C^{\\prime}B^{\\prime}$\r\n- é©åœãªè§åºŠèšç®ãããããšã«ãã£ãŠ $\\triangle BHC \\equiv \\triangle CTB^{\\prime}$\r\n- ååãçšããŠïŒ$AT=\\sqrt{2}c, A^{\\prime}T=\\sqrt{2}b, \\angle ATA^{\\prime}=135^{\\circ}$\r\n\r\nã$AA^{\\prime}=2OA=14$ ãçšããã°ïŒãã¯ãäœåŒŠå®çã掻çšããããšãã§ããïŒ",
"text": "äžç·å®çã䜿ããªãæ¹é",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/10611/569"
},
{
"content": "ã$BE = a, EC = EA = b$ ãšããïŒãã®ãšãïŒåè§åœ¢ $BECO$ ã $4$ ã€çµã¿åãããããšã§äžèŸºã®é·ãã $a+b$ ã®æ£æ¹åœ¢ãã§ããã®ã§ïŒåè§åœ¢ $BECO$ ã®é¢ç©ã¯ $\\dfrac{(a+b)^2}{4}$ ãšåããïŒãŸãïŒ$AB = AE + EB = a + b$ ããïŒäžè§åœ¢ $ADB$ ã®é¢ç©ã $\\dfrac{(a+b)^2}{4}$ ãšåããïŒãã£ãŠïŒåè§åœ¢ $BECO$ ã®é¢ç©ãšäžè§åœ¢ $ADB$ ã®é¢ç©ãçããããïŒåè§åœ¢ $AEHD$ ã®é¢ç©ãšåè§åœ¢ $BHCO$ ã®é¢ç©ã¯çããïŒå
±ã« $5$ ãšåããïŒ\\\r\nãããã§ïŒäžè§åœ¢ $ABC$ïŒäžè§åœ¢ $BOC$ïŒäžè§åœ¢ $BHC$ ã®é¢ç©ããããã $S, T, U$ ãšããïŒãããšïŒ$T + U = 5$ ãçŽã¡ã«åããïŒãŸãïŒ$AD : AB = AE : AC = 1 : \\sqrt{2}$ ããïŒäžè§åœ¢ $ADE$ ãšäžè§åœ¢ $ABC$ ã¯çžäŒŒæ¯ $1 : \\sqrt{2}$ïŒã€ãŸãïŒé¢ç©æ¯ $1 : 2$ ã®çžäŒŒãªäžè§åœ¢å士ã§ãããšåããïŒåæ§ã«ïŒ$EH : BH = DH : CH = 1: \\sqrt{2}$ ããïŒäžè§åœ¢ $EHD$ ãšäžè§åœ¢ $BHC$ ã®é¢ç©æ¯ã $1 : 2$ ãšãåããïŒãã£ãŠïŒåè§åœ¢ $AEHD$ ã®é¢ç©ã¯ $\\dfrac{S + U}{2}$ ãšãªãïŒ$S + U = 10$ ãåããïŒ\\\r\nãããã§ïŒäžè§åœ¢ $AOC$ ãç¹ $O$ ãäžå¿ã«ç¹ $C$ ãç¹ $B$ ã«éãªãããã«å転移åãããããšãèããïŒãã®ãšãïŒç¹ $A$ ã®ç§»åå
ã $A^\\prime$ ãšããïŒãããšïŒäžè§åœ¢ $AOA^\\prime$ 㯠$AO = A^\\prime O = 7, \\ \\angle AOA^\\prime = 90^\\circ$ ã®çŽè§äºç蟺äžè§åœ¢ã«ãªãã®ã§ïŒãã®é¢ç©ã¯ $\\dfrac{49}{2}$ ãšãªãïŒãŸãïŒåè§åœ¢ $AOA^\\prime B$ ã®é¢ç©ã¯åè§åœ¢ $ABOC$ ã®é¢ç©ã«çããïŒ$S + T$ ãšè¡šãããïŒãããŠïŒäžè§åœ¢ $ABA^\\prime$ 㯠$A^\\prime B = AC, \\ \\angle ABA^\\prime = 135^\\circ$ ãæºããããïŒãã®é¢ç©ã¯äžè§åœ¢ $ABC$ ã®ããã«çããïŒ$S$ ã§è¡šãããïŒãã£ãŠïŒ$2S + T = \\dfrac{49}{2}$ ãåŸãïŒ\\\r\nã以äžãé£ç«ãããŠè§£ãããšã§ïŒ$T = \\dfrac{29}{6}$ ãåŸãããïŒ$T = \\dfrac{BC^2}{4}$ ããïŒ$BC^2 = \\dfrac{58}{3}$ ãåŸãïŒ",
"text": "é¢ç©ã«é¢ããé£ç«æ¹çšåŒãç«ãŠãŠè§£ãæ¹æ³",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/10611/570"
},
{
"content": "ã$BC=2x$ ãšããïŒ \\\r\nã蟺 $BC$ ã®äžç¹ã $M$ ïŒäžè§åœ¢ $ABC$ ã®å€å¿ã $O^\\prime$ ãšãããšïŒ$OM=OâM=x$ ãš $AO=\\sqrt{2} x$ ããäžç·å®çã§\\\r\n$$7^2+(\\sqrt{2} x)^2=2(x^2+AM^2)$$\r\n$$AM=\\frac{7\\sqrt{2}}{2}$$ ããããïŒ\\\r\nãäžç·å®çããïŒ$AB^2+AC^2=2(AM^2+BM^2)=49+2x^2$ \\\r\nããŸãïŒ$AE=EC,AD=DB$ ãã $AE^2+EB^2+AD^2+DC^2=EC^2+EB^2+DB^2+DC^2=2BC^2=8x^2$ \\\r\nã$BE=EH,CD=DH$ ãã \r\n$$\\begin{aligned}\r\nAB^2+AC^2-(AE^2+EB^2+AD^2+DC^2) & =(AE+EB)^2+(AD+DC)^2-(AE^2+EB^2+AD^2+DC^2) \\\\\\\\\r\n& =2AEã»EB+2ADã»DC \\\\\\\\\r\n& =2AEã»EH+2ADã»DH \\\\\\\\\r\n& =20\r\n\\end{aligned}$$\r\nããã£ãŠ $(49+2x^2)-8x^2=20$ ãªã®ã§\r\n$$\\begin{aligned}\r\nBC^2 & = (2x)^2 \\\\\\\\\r\n& =\\frac{58}{3}\r\n\\end{aligned}$$\r\nãªã®ã§ïŒè§£çãã¹ãå€ã¯ $\\mathbf{61}$ ïŒ",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/10611/575"
}
] | ã$\angle A = 45^\circ$ ã§ãããããªéè§äžè§åœ¢ $ABC$ ã®åå¿ã $H$ ãšããŸãïŒ$B, C$ ãã察蟺ã«äžãããåç·ã®è¶³ããããã $D, E$ ãšãïŒäžè§åœ¢ $BHC$ ã®å€å¿ã $O$ ãšãããšïŒ$AO = 7$ ã§ããïŒãã€åè§åœ¢ $AEHD$ ã®é¢ç©ã $5$ ãšãªããŸããïŒãã®ãšãïŒ$BC^{2}$ ã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠäžããïŒ |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/9342 | C | OMC225(C) | 400 | 75 | 124 | [
{
"content": "ãæ±ããã¹ãã¯ïŒ$100 - 4x \\lt z + 2y \\lt 100 + 4x$ ã〠$4x - 100 \\lt 2y - z \\lt 100 - 4x$ ãæºããéè² æŽæ° $(x,y,z)$ ã®çµã®æ°ã§ããïŒãã㧠$x$ ã®å€ãåºå®ããŠïŒæ¡ä»¶ãæºããç¹ $(y, z)$ ã®é åã $yz$ 座æšå¹³é¢äžã«å³ç€ºããããšãèããïŒ$4x - 100 \\lt 100 - 4x$ ãã $0 \\lt x \\lt 25$ ã®å Žåã®ã¿èããã°ããïŒãã®ãšãç¹ $(y, z)$ ã®é å㯠$4$ çŽç·\r\n$$z = -2y + 100 - 4x, \\quad z = -2y + 100 + 4x \\\\\\\\\r\nz = 2y + 100 - 4x, \\quad z = 2y + 4x - 100$$\r\nã«å²ãŸããå
éšé å (å¢çãé€ã) ãšãªãïŒãã㯠$4$ ç¹ \r\n$$(x,0, 100 - 4x) , \\ (x,50 - 2x, 0) , \\ (x,2x, 100) , \\ (x,50, 4x)$$\r\nãé ç¹ã«æã€å¹³è¡å蟺圢ã§ããããšãåããïŒãã®å¹³è¡å蟺圢ã®é¢ç©ã¯ \r\n$$50 \\times 100 - \\left (2x \\times 4x \\times \\cfrac{1}{2} + (50 - 2x) \\times (100 - 4x) \\times \\cfrac{1}{2} \\right ) \\times 2 = 400x - 16x^{2}$$\r\nã§ããïŒãŸã蟺äžã«ããæ Œåç¹ã®æ°ã¯ïŒé ç¹ä»¥å€ã $(2x - 1 + 50 - 2x - 1) \\times 2 = 96$ åïŒé ç¹ã $4$ åã§ããããïŒåãã㊠$100$ åãšåããïŒä»¥äžããïŒããã¯ã®å®çããïŒå¢çãé€ãé åå
ã®æ Œåç¹ã®åæ°ã¯ $400x - 16x^{2} - 49$ åãšåããïŒæ±ããã¹ãå€ã¯ããã $1 \\leq x \\leq 24$ ã®ç¯å²å
ã§è¶³ãåãããå€ã§ããïŒèšç®ãããšçã㯠$\\mathbf{40424}$ åãšãªãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/9342"
}
] | ãéè² æŽæ°ã®çµ $(x, y, z)$ ã§ãã£ãŠïŒ
$$\lvert 100 - 4x - 2y \rvert \lt z \lt 100 - \lvert 4x - 2y \rvert$$
ãæºãããã®ã¯ããã€ãããŸããïŒ |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/8064 | D | OMC225(D) | 400 | 97 | 172 | [
{
"content": "ãç·å $AC$ ã $4$ çåããç¹ãïŒ$A$ ã«è¿ãæ¹ãã $A_1,A_2,A_3$ ãšãïŒ$A_4=C$ ãšããïŒ$A$ ãã $A_i$ ãŸã§ç·ã®äžã®ã¿ãéã£ãŠæçã§ç§»åããæ¹æ³ã®ç·æ°ã $f(i)$ ã§è¡šãïŒ$f(4)$ ãæ±ããã°ããïŒ\\\r\nããŸãïŒ$A_1$ ãéãæ¹æ³ã¯ïŒ$A$ ãã $A_1$ ãŸã§ã $f(1)$ éãïŒ$A_1$ ãã $C$ ãŸã§ã $f(3)$ éãã§ïŒãããã¯ç¬ç«ãªã®ã§å
šäœã§ã¯ $f(1)f(3)$ éãã§ããïŒæ¬¡ã«ïŒ$A_1$ ãéãã $A_2$ ãéãæ¹æ³ã¯ïŒåæ§ã«èã㊠$2f(2)$ éãã§ããïŒããã«ç¹°ãè¿ãããšã§ïŒ$f(4)=f(1)f(3)+2f(2)+4f(1)+10$ ãæãç«ã€ããšããããïŒ\\\r\nãåæ§ã«é¡ã£ãŠããããšã§ïŒ$f(3)=f(1)f(2)+2f(1)+4$ïŒ$f(2)=f(1)^2+2$ ã§ããããïŒãŸãšãããš\r\n$$ f(4) = f(1)^4 + 6f(1)^2 + 8f(1) + 14.$$\r\nã以äžïŒ$f(1)$ ãæ±ããïŒç·å $AA_1$ ã®äžç¹ã $X$ ãšãïŒ$A$ ãã $X$ ãŸã§ã®æççµè·¯ã $N$ éããããšãããšïŒäžãšåæ§ã«ã㊠$f(1)=N^2+2$ ã§ããïŒããã«ïŒç·å $AX$ ã®äžç¹ã $Y$ ãšãïŒ$A$ ãã $Y$ ãŸã§ã®æççµè·¯ã $M$ éããããšãããšïŒãã¯ã $N=M^2+2$ ã§ããïŒ$M={}\\_{4}\\mathrm{C}_2=6$ ã§ããããïŒé ã«èšç®ããããšã§ $f(4)=\\mathbf{4371942276134}$ ãåŸãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/8064"
}
] | ãæ£æ¹åœ¢ $ABCD$ ãããïŒå蟺ã $4$ çåãããããªç¹ã«ãã£ãŠ $16~(=4\times 4)$ åã®å°æ£æ¹åœ¢ã«åãããããã«ç·ãåŒããŸãïŒããã«ïŒä»¥äžã®æäœã $3$ åè¡ããŸãïŒããã§ïŒåæç¹ã§ãå°æ£æ¹åœ¢ããšãã£ããšãïŒãã®å
éšïŒåšäžãé€ãïŒã«ãããªãç·ãåŒãããŠããªããã®ããããã®ãšããŸãïŒ
- ç·å $AC$ ãå
éšïŒåšäžãé€ãïŒãéããããªå°æ£æ¹åœ¢ãããããïŒå蟺ã®äžç¹ã«ãã£ãŠ$4~(=2\times 2)$ åã®å°æ£æ¹åœ¢ã«åãããããã«ç·ãåŒãïŒ
æçµçãªç¶æ³ã«ãããŠïŒç·ã®äžã®ã¿ãéã£ãŠ$A$ ãã $C$ ãŸã§æçã§å°éããæ¹æ³ã®ç·æ°ãæ±ããŠãã ããïŒ
<details><summary>$1$ åç®ã®æäœãçµäºããæç¹ã®å³<\/summary>
<\/details>
<details><summary>$2$ åç®ã®æäœãçµäºããæç¹ã®å³<\/summary>
<\/details>
<details><summary>ãã¹ãŠã®æäœãçµäºããæç¹ã®å³<\/summary>
<\/details> |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/9337 | E | OMC225(E) | 500 | 20 | 52 | [
{
"content": "ã$ζ$ ã $1$ ã®åå§ $101$ ä¹æ ¹ $\\left(= \\cos \\dfrac{2\\pi}{101}+ i\\sin\\dfrac{2\\pi}{101}\\right)$ ãšããïŒãã®ãšãïŒ\r\n$$X^{100} + X^{99} + \\cdots + X + 1 = (X - ζ) (X - ζ^{2}) \\cdots (X - ζ^{100})$$\r\nãæãç«ã€ïŒãã®åŒã $(1)$ ãšããïŒãã®ãšãïŒæ±ããã¹ãå€ã¯ \r\n$$\\prod_{i = 1}^{101} (α_{i} - ζ) (α_{i} - ζ^{2}) \\cdots (α_{i} - ζ^{100})$$\r\nãšãªãïŒãã㧠$f(X) = X^{101} + 2024X^{50} - 2025$ ãšãããšïŒå æ°å®çãã\r\n$$f(X) = (X - α_{1}) (X - α_{2}) \\cdots (X - α_{101})$$\r\nãæãç«ã€ããïŒæ±ããã¹ãå€ã $\\displaystyle \\prod_{i = 1}^{100} \\bigl(-f(ζ^{i})\\bigr) = \\displaystyle \\prod_{i = 1}^{100} f(ζ^{i})$ ã§ããããšãåããïŒ$ζ^{101} = 1$ ã«çæãããšïŒ\r\n$$ f(ζ^{i}) = 2024 \\ ( {ζ^{50i}} - 1 )$$\r\nãåŸãããïŒæ±ããã¹ãå€ã¯ \r\n$$ \\prod_{i = 1}^{100} 2024 \\ ( ζ^{50i} - 1 ) = 2024^{100} \\ \\prod_{i = 1}^{100} (ζ^{50i} - 1) = 2024^{100} \\ \\prod_{i = 1}^{100} (ζ^{i} - 1)$$\r\nãšãªãïŒ$(1)$ ã« $X = 1$ ã代å
¥ããããšã§ïŒ$\\displaystyle \\prod_{i = 1}^{100} (ζ^{i} - 1) = 101$ ãåŸãããïŒæçµçã«æ±ããã¹ãå€ã¯\r\n$$2024^{100} à 101 = 2^{300} à 11^{100} à 23^{100} à 101$$\r\nãšãªãïŒãããæã€æ£ã®çŽæ°ã®åæ°ã¯ $\\mathbf{6141002}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/9337"
},
{
"content": "ãæ¹çšåŒ $X^{101}+2024X^{50}-2025=0$ 㯠$X=1$ ã解ã«æã€ïŒ\r\n$$X^{101}+2024X^{50}-2025=(X-1) \\lbrace (X^{100}+\\cdots+1)+2024(X^{49}+\\cdots+1) \\rbrace$$\r\nã§ããïŒ$X=1$ ã¯é解ã§ã¯ãªãïŒãã㧠$\\alpha_{101}=1$ ãšãããïŒããŸæ±ããããã®ã¯\r\n\r\n$$\\begin{aligned}\r\n\\prod_{i=1}^{101} \\left( \\sum_{j=0}^{100}(\\alpha_i)^j\\right) &= \\prod_{i=1}^{100} \\left( \\sum_{j=0}^{100}(\\alpha_i)^j\\right) Ã\\sum_{j=0}^{100}(\\alpha_{101})^j\\\\\\\\\r\n& = 101Ã\\prod_{i=1}^{100}\\dfrac{1-\\alpha_i ^{\\ 101}}{1-\\alpha_i} \\\\\\\\\r\n& = 101Ã\\dfrac{\\prod\\limits_{i=1}^{100} (1-\\alpha_i ^{\\ 101})}{\\prod\\limits_{i=1}^{100}(1-\\alpha_i)}\r\n\\end{aligned}$$\r\nãšå€åœ¢ãããïŒãã®æåŸã®åŒã®åæ¯ã¯ïŒ\r\n$$\\prod_{i=1}^{100}(X-\\alpha_i)=(X^{100}+\\cdots+1)+2024(X^{49}+\\cdots+1)$$\r\nãçšããã°ïŒ$\\prod\\limits_{i=1}^{100}(1-\\alpha_i)=101+2024Ã50$ ã§ããïŒ\\\r\nãããšã¯å€é
åŒ $\\prod\\limits_{i=1}^{100} (X-\\alpha_i ^{\\ 101})$ ãæ±ããããšãã§ããã°ããïŒ\r\n\r\n---\r\n\r\nãããã§å
ã®æ¹çšåŒã®ææ°ãé©åœã«å€ããŠïŒåŒ\r\nã$$X+2024X^\\frac{50}{101}-2025=0$$\r\n㯠$\\alpha_1^{\\ 101} \\cdots \\alpha_{101}^{\\ \\ 101}$ ã解ã«æã€ïŒãã®æ¹çšåŒãé©åœã«å€åœ¢ããŠ\r\nã$$2024^{101}X^{50}+(X-2025)^{101}=0$$\r\nãšããã°ïŒãã®åŒã¯ $\\alpha_1^{\\ 101} \\cdots \\alpha_{101}^{\\ \\ 101}$ ã解ã«æã€å€é
åŒãšãªãïŒ\r\n<details><summary>ããæ£ç¢ºã«ã¯<\\/summary>\r\n\r\nã$\\alpha_1^{\\ 101} \\cdots \\alpha_{101}^{\\ \\ 101}$ ãçžç°ãªãã°ååã§ããïŒæ¹çšåŒ \r\nã$$ \\tag{1} 2024^{101}X^{50}+(X-2025)^{101}=0$$\r\nãé解ãæã€ãšä»®å®ããã°ïŒãã®æ¹çšåŒã埮åãã次ã®æ¹çšåŒ\r\nã$$ \\tag{2} 50Ã2024^{101}X^{49}+101(X-2025)^{100}=0$$\r\nãåã解ãæã€ããšã«ãªãïŒ$101à (1) - (X-2025)Ã(2)$ ãèšç®ããã° $X$ ããã äžã€ã«å®ãŸãïŒãã®å€ã¯åŒ $(1)$ ãæºãããªãïŒ\r\n\r\n<\\/details>\r\n\r\nãæ®ããã課é¡ã¯ïŒå€é
åŒ $2024^{101}X^{50}+(X-2025)^{101}$ ã $(X-1)P(X)$ ã®åœ¢ã«å€åœ¢ããŠïŒ$P(1)$ ãæ±ããããšã§ããïŒ\r\n$$\\begin{aligned}\r\n2024^{101}X^{50}+(X-2025)^{101} &= 2024^{101}X^{50}+\\lbrace (X-1)-2024 \\rbrace^{101} \\\\\\\\\r\n& = 2024^{101}X^{50}+(X-1)^2Q(X)+101Ã2024^{100}(X-1)-2024^{101}\\\\\\\\\r\n& = (X-1)^2Q(X)+101Ã2024^{100}(X-1)+2024^{101}(X-1)(X^{49}+\\cdots+1)\\\\\\\\\r\n& = (X-1) \\lbrace (X-1)Q(X)+101Ã2024^{100}+2024^{101}(X^{49}+\\cdots+1) \\rbrace\r\n\\end{aligned}$$\r\nïŒäžã®åŒå€åœ¢ã«ãããŠïŒäºã€ç®ã®çå·ã§ã¯äºé
å®çãçšããïŒïŒ\\\r\nããã£ãŠ $P(1)=101Ã2024^{100}+2024^{101}Ã50=2024^{100}Ã(101+2024Ã50)$ ãšãªãïŒä»¥äžã®èšç®ããïŒ\r\n$$\\prod_{i=1}^{101} \\left( \\sum_{j=0}^{100}(\\alpha_i)^j\\right) = 101Ã\\dfrac{\\prod\\limits_{i=1}^{100} (1-\\alpha_i ^{\\ 101})}{\\prod\\limits_{i=1}^{100}(1-\\alpha_i)}=101Ã\\dfrac{2024^{100}Ã(101+2024Ã50)}{101+2024Ã50}=101Ã2024^{100}$$",
"text": "åå§101ä¹æ ¹ã䜿ããªãæ¹é",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/9337/571"
}
] | ã$X$ ã«é¢ãã $101$ 次æ¹çšåŒ
$$X^{101} + 2024X^{50} - 2025 = 0$$
ã®ïŒéè€åºŠã蟌ããŠïŒ$101$ åã®è€çŽ æ°è§£ã $X=α_{1}, α_{2}, \ldots , α_{101}$ ãšããŸãïŒãã®ãšãïŒ
$$\prod_{i = 1}^{101} \left (\sum_{j = 0}^{100} (α_{i})^{j} \right )$$
ã¯æ£æŽæ°å€ã«ãªãã®ã§ïŒããããã€æ£ã®çŽæ°ã®åæ°ã解çããŠãã ããïŒ |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/9208 | F | OMC225(F) | 500 | 40 | 66 | [
{
"content": "**è£é¡1.**ã$2$ 以äžã®æŽæ° $n$ ã«å¯ŸãïŒ$ \\ a_{2n} = 2a_{2n-2} + a_{2n-1}, \\ a_{2n+1} = a_{2n-1} + a_{2n}$ ãæç«ããïŒ\r\n\r\n**蚌æ.**ã$\\lbrace a_{n} \\rbrace$ ã®å®ãæ¹ããïŒ\r\n$$\\begin{aligned}\r\na_{2n} &= (a_{1} + a_{2} + \\cdots + a_{2n-3}) + a_{2n-2} + a_{2n-1} = 2a_{2n-2} + a_{2n-1}, \\\\\\\\\r\na_{2n+1} &= (a_{2} + a_{4} + \\cdots + a_{2n-2}) + a_{2n} = a_{2n-1} + a_{2n}\r\n\\end{aligned}$$\r\nã $n \\geq 2$ ã«å¯ŸããŠæç«ããïŒâ \r\n\r\n---\r\n\r\n**è£é¡2.**ã$6$ 以äžã®æŽæ° $n$ ã«å¯ŸãïŒ$a_{n} = 4a_{n-2} - 2a_{n-4}$ ãæç«ããïŒ\r\n\r\n**蚌æ.**ã$n$ ãå¶æ°ã®ãšãïŒè£é¡1ãã\r\n$$a_{n} = 2a_{n-2} + a_{n-1} = a_{n-3} + 3a_{n-2} = 4a_{n-2} - 2a_{n-4}$$\r\nã $n \\geq 6$ ã«å¯ŸããŠæç«ããïŒãŸãïŒ$n$ ãå¥æ°ã®ãšãïŒè£é¡1ãã\r\n$$a_{n} = a_{n-2} + a_{n-1} = 2a_{n-3} + 2a_{n-2} = 4a_{n-2} - 2a_{n-4}$$\r\nã $n \\geq 7$ ã«å¯ŸããŠæç«ããïŒâ \r\n\r\n---\r\n\r\n**è£é¡3.**ãä»»æã®æ£æŽæ° $n$ ã«å¯ŸãïŒ$a_{n}$ ãš $a_{n+1}$ ããšãã«å²ãåãå¥çŽ æ°ã¯ååšããªãïŒ\r\n\r\n**蚌æ.**ã$a_{n}$ ãš $a_{n+1}$ ãå
±éã®å¥çŽ å æ° $p$ ããã€ãããªïŒæå°ã® $n$ ããšãïŒ$a_1=a_2=1$ ã«ãã $n\\geq 2$ ã§ããïŒãããšïŒ$n$ ãå¶æ°ã®ãšã㯠$ a_{n-1} = a_{n+1} - a_{n}$ ãïŒ$n$ ãå¥æ°ã®ãšã㯠$2a_{n-1} = a_{n+1} - a_{n}$ ã $p$ ã§å²ãåããããšã«ãªãïŒãããã«ããïŒ$p$ ãå¥æ°ã§ããããšããïŒ$a_{n-1}$ ã $p$ ã§å²ãåããããïŒæå°æ§ã«ççŸããïŒâ \r\n\r\n---\r\n\r\nãè£é¡1ãšè£é¡3ããïŒãä»»æã®æ£æŽæ° $n$ ã«å¯ŸããŠïŒ$a_{n}$ ãš $a_{n+2}$ ããšãã«å²ãåãå¥çŽ æ° $p$ ã¯ååšããªããããšã瀺ãããïŒããã«ïŒãã®äºå®ãšè£é¡2ããïŒãä»»æã®æ£æŽæ° $n$ ã«å¯ŸããŠïŒ$a_{n}$ ãš $a_{n+4}$ ããšãã«å²ãåãå¥çŽ æ° $p$ ã¯ååšããªããããšã瀺ãããïŒããããïŒ$\\gcd(a_{n}, a_{n+4})$ 㯠$2$ åªã«ãªãïŒãã£ãŠïŒä»¥äž $a_{n}$ ã $2$ ã§å²ãåããæ倧ã®åæ° $b_n$ ã«ã€ããŠèããïŒ\\\r\nãè£é¡2ãå©çšããããšã§ïŒ$b_{1} = b_{2} = b_{3} = b_{4} = 0$ ããã³ $5$ 以äžã®æ·»åã«å¯Ÿã\r\n$$\\begin{aligned}\r\nb_{8k} &= 2k - 1, & b_{8k+1} &\\geq 2k + 2, & b_{8k+2} &= 2k, & b_{8k+3} &= 2k, \\\\\\\\\r\nb_{8k+4} &= 2k, & b_{8k+5} &= 2k + 2, & b_{8k+6} &= 2k+1, & b_{8k+7} &= 2k+1\r\n\\end{aligned}$$\r\nãæãç«ã€ããšãåããïŒããããïŒ$\\gcd(a_{n}, a_{n+4})=2^{c_{n}}$ ãšããã°ïŒ\r\n$$\\begin{aligned}\r\nc_{8k} &= 2k-1, & c_{8k+1} &= 2k+2, & c_{8k+2} &= 2k, & c_{8k+3} &= 2k, \\\\\\\\\r\nc_{8k+4} &= 2k, & c_{8k+5} &= 2k + 2, & c_{8k+6} &= 2k + 1, & c_{8k+7} &= 2k + 1\r\n\\end{aligned}$$\r\nãæãç«ã€ïŒ$2^{49} \\lt 10^{15} \\lt 2^{50}$ ã«ããïŒæ±ããæå°å€ã¯ $n = \\mathbf{193}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/9208"
},
{
"content": "ãå
¬åŒè§£èª¬ã¯ $n$ ã®å¶å¥ã«ãã£ãŠåããŠããªããã®ã®ïŒæŒžååŒã®åœ¢ãã $n$ ã®å¶å¥ã§åããŠèããã®ãèªç¶ãªçºæ³ã§ããïŒããã§ã¯ïŒãã®ãããªæ¹éã§äœæãã解çãèšããŠããïŒ\\\r\nããªãïŒå
¬åŒè§£èª¬ãšæ¬è³ªçã«å€§ããå€ããããã§ã¯ãªãïŒä»¥äžïŒå
¬åŒè§£èª¬ã®è£é¡ 1,2,3 ãšãããã察å¿ãããªããèšè¿°ããŠããïŒ\r\n\r\nããŸãæ°å $\\lbrace a_n \\rbrace$ ãå¥æ°éšåãšå¶æ°éšåã«åããïŒ$a_{2n-1}=s_n, a_{2n}=t_n$ ãšãããïŒ\r\n\r\n---\r\n\r\nã**è£é¡ 1.**ã$2$ 以äžã®æŽæ° $n$ ã«å¯ŸãïŒ$s_{n+1}=s_n+t_n, t_{n+1}=s_n+3t_n$ ãæç«ããïŒïŒèšŒæç¥ïŒ\r\n\r\n---\r\n\r\nã**è£é¡ 2.**ã$\\gcd (s_n,s_{n+2})=\\gcd(s_n, 4t_n)$ïŒ$\\gcd (t_n,t_{n+2})=\\gcd(t_n, 4s_n)$\r\n\r\nã**蚌æ.**ãè£é¡ 1 ã®æŒžååŒãçšããŠïŒ$s_{n+2}=2s_n+4t_n, t_{n+2}=4s_n+10t_n$ ãåŸãïŒããšã¯ Euclid ã®äºé€æ³ãçšããã°ããïŒ\r\n\r\n---\r\n\r\nã**è£é¡ 3.**ãä»»æã®æ£æŽæ° $n$ ã«å¯ŸããŠïŒ$s_n$ ãš $t_n$ ããšãã«å²ãåãå¥çŽ æ°ã¯ååšããªãïŒ\r\n\r\nã**蚌æ.**ã\r\n$$\\gcd (s_{n+1},t_{n+1})=\\gcd(s_n+t_n, s_n+3t_n)=\\gcd(s_n+t_n,2t_n) \\leq \\gcd(2s_n+2t_n,2t_n) =\\gcd(2s_n,2t_n)$$\r\nïŒéäžã®äžçå·ã¯ïŒåã«å€§å°é¢ä¿ã ãã§ã¯ãªãïŒäžæ¹ãä»æ¹ã®çŽæ°ã§ããããšãå«æããŠããïŒïŒ\\\r\nããã®èšç®ããïŒããå¥çŽ æ° $p$ ã«ã€ã㊠$p \\mid \\gcd (s_{n+1},t_{n+1})$ ãæãç«ãŠã° $p \\mid \\gcd (s_{n},t_{n})$ ãæãç«ã€ïŒ$\\gcd (s_2,t_2)=1$ ããïŒãã®ãããªå¥çŽ æ° $p$ ã¯ååšããªãïŒ\r\n\r\n---\r\n\r\nãè£é¡ 2 ãšè£é¡ 3 ãã $\\gcd (s_n,s_{n+2}), \\gcd (t_n,t_{n+2})$ ã¯ãšãã« $2$ åªã§ããïŒãã£ãŠïŒä»¥äž $s_n,t_n$ ã $2$ ã§å²ãåããæ倧ã®åæ°ãèããïŒ\\\r\nããã㧠$n=2 \\sim 6$ ã«ã€ããŠïŒè£é¡ 1 ã®æŒžååŒãçšããªããïŒ$s_n,t_n$ ã $32$ ã§å²ã£ãäœããæžã䞊ã¹ãŠã¿ããïŒ\r\n$$\\begin{matrix}\r\n \\ & n=2 & n=3 & n=4 & n=5 & n=6 \\cr\r\n s_n & 1 & 4 & 14 & 16 & 4 \\cr\r\n t_n & 3 & 10 & 2 & 20 & 12 \\cr\r\n\\end{matrix}$$\r\nã$(s_6, t_6)$ ã $(s_2,t_2)$ ã®ã¡ããã© $4$ åã«ãªã£ãŠããããšããããïŒãã®ããšããçæ³ãåŸãŠïŒ$n=4k+2 \\sim 4k+6$ ã«ã€ããŠïŒ$s_n,t_n$ ã $2^{2k+5}$ ã§å²ã£ãäœãã次ã®ããã«åŸãããïŒ\r\n\r\n$$\\begin{matrix}\r\n \\ & n=4k+2 & n=4k+3 & n=4k+4 & n=4k+5 & n=4k+6 \\cr\r\n s_n & (8x+1)2^{2k} & \\lbrace 8(x+y)+4 \\rbrace 2^{2k} & (16x+14)2^{2k} & \\lbrace 16(x+y)+16 \\rbrace2^{2k} & 4Ã2^{2k} \\cr\r\n t_n & (8y+3)2^{2k} & \\lbrace 8(x+3y)+10 \\rbrace 2^{2k} & (16y+2)2^{2k} & \\lbrace 16(x+y)+20 \\rbrace 2^{2k} & 12Ã2^{2k} \\cr\r\n\\end{matrix}$$\r\n\r\nããã®è¡šãèŠãã°ïŒ$4k+2 \\leq n \\leq 4k+6$ ã®ç¯å²ã§ $\\gcd(s_n, 4t_n)$ ãŸã㯠$\\gcd(t_n, 4s_n)$ ã®æ倧å€ã $\\gcd(s_{4k+5}, 4t_{4k+5})=2^{2k+4}$ ã§ãããšãããïŒ\\\r\nã$2^{49} \\lt 10^{15} \\lt 2^{50}$ ããïŒ$2k+4=50$ ãã $k=23$ ïŒ$s_{97}=a_{193}$ ããæ±ããã¹ãå€ã¯ $\\mathbf{193}$ ã§ããïŒ",
"text": "n ã®å¶å¥ã§åããå Žå",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/9208/568"
},
{
"content": "$$f(x)=a_1+a_3x+a_5x^2+\\dots,\\quad g(x)=a_2+a_4x+a_6x^2+\\dots$$\r\nãšãã.ãã®ãšã, \r\n$$\r\n\\begin{aligned}\r\nf(x)-\\dfrac{x}{1-x}g(x)&=a_1+(a_3-a_2)x+(a_5-a_4-a_2)x^2+\\dots\\\\\\\\\r\n&=1\\\\\\\\\r\ng(x)-\\dfrac{1}{1-x}f(x)-\\dfrac{x}{1-x}g(x)&=(a_2-a_1)+(a_4-a_3-a_2-a_1)x+(a_6-a_5-a_4-a_3-a_2-a_1)x^2+\\dots\\\\\\\\\r\n&=0\r\n\\end{aligned}$$\r\n第äºåŒãã, $f=(1-2x)g$ ã§ãããã, ããã第äžåŒã«ä»£å
¥ããããšã§,\r\n$$\r\n\\begin{aligned}\r\n&f(x)=\\frac{(1-2x)(1-x)}{1-4x+2x^2}\\\\\\\\\r\n&g(x)=\\frac{1-x}{1-4x+2x^2}\r\n\\end{aligned}$$\r\nãæç«. ãã£ãŠ,$f(x),g(x)$ ããããã« $1-4x+2x^2 $ ãããããšãããã $3,2$ 次以éã®é
ã $0$ ã«ãªãããšãã $n\\geq 6$ ã«ã€ããŠ, $a_n-4a_{n-2}+2a_{n-4}=0$ ãåãã.",
"text": "挞ååŒãæ¯é¢æ°ãçšããŠå°åº",
"url": "https://onlinemathcontest.com/contests/omc225/editorial/9208/631"
}
] | ãæ£æŽæ°å $\lbrace a_{n} \rbrace\_{n=1,2,\ldots}$ ã以äžã®ããã«å®ããŸãïŒ
- $a_{1} = 1$ïŒ
- $n$ ãå¶æ°ã®ãšãïŒ$a_{n} = a_{1} + a_{2} + a_{3} + \cdots + a_{n-1}$ïŒ
- $n$ ãå¥æ°ã®ãšãïŒ$a_{n} = a_{2} + a_{4} + a_{6} + \cdots + a_{n-1}$ïŒ
ããã®ãšãïŒ$\gcd(a_{n}, a_{n+4}) \geq 10^{15}$ ãã¿ããæå°ã®æ£æŽæ° $n$ ãæ±ããŠãã ããïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/4432 | A | OMCB016(A) | 100 | 319 | 330 | [
{
"content": "ã$n!$ ã $10$ ã§ã¡ããã© $10$ åå²ãåããããšãæ¡ä»¶ã§ããïŒããã¯ããã« $n!$ ã $5$ ã§ã¡ããã© $10$ åå²ãåãããšèšãæããŠããïŒ$n!$ ã $5$ ã§å²ãåããåæ°ã¯å調å¢å ã§ããããšã«æ°ãã€ãããšïŒ$44!,45!,\\ldots,49!,50!$ ã¯ãããã $5$ ã§ã¡ããã© $9,10,\\ldots,10,12$ åå²ãåããããšããïŒæ±ããç·å㯠$45+\\cdots+49=\\mathbf{235}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb016/editorial/4432"
}
] | ã$n!$ ã®åé²æ³è¡šèšã«ãããŠïŒæ«å°Ÿã« $0$ ãã¡ããã© $10$ å䞊ã¶ãããªæ£æŽæ° $n$ ã®ç·åãæ±ããŠãã ãã. |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/8193 | B | OMCB016(B) | 200 | 91 | 198 | [
{
"content": "ã$Q(x)=P(x)-(x-11)^2$ ãšãããšïŒ$Q(x)$ ããŸãæŽæ°ä¿æ°å€é
åŒã§ããïŒ$Q(3)=Q(8)=Q(13)=0$ ãã¿ããïŒãããã£ãŠïŒå æ°å®çã«ããããæŽæ°ä¿æ°å€é
åŒ $R(x)$ ãååšã㊠$Q(x)=(x-3)(x-8)(x-13)R(x)$ ãã¿ããïŒãã®ãšã $P(10)=1-42R(10)$ ãšãªãããïŒ$P(10)$ ã®åãåŸãå€ã¯ $42$ ã§å²ã£ãäœãã $1$ ã§ããæŽæ°å
šäœã§ããïŒååæ§ã¯ $R(x)$ ãå®æ°ãšããããšã§ãããïŒïŒãã£ãŠïŒè§£çãã¹ãå€ã¯\r\n$$\\sum_{k=0}^{23}(42k+1)=\\mathbf{11616}.$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb016/editorial/8193"
},
{
"content": "ããã®æã®åé¡ã¯\r\n$$\\tag{1} P(3)=64, P(8)=9, P(13)=4$$\r\nãæºããå€é
åŒãäžã€äœãã°ïŒããšã¯ãããåºæºã«ããŠãã¹ãŠã®å€é
åŒãè¡šããå Žåãå€ãã§ãïŒ\\\r\nããã£ãŠïŒæåã«ãã¹ãããšã¯åŒ $(1)$ ãæºããå€é
åŒããšã«ããäžã€äœã£ãŠããŸãããšããããšã«ãªããŸãïŒ\\\r\nããã®ããã®æ¹æ³ãšããŠïŒ$f(x)=ax^2+bx+c$ ãšãããŠãã\r\n$$f(3)=64, f(8)=9, f(13)=4$$\r\nã解ãã®ãç«æŽŸãªæ¹æ³ã§ãïŒ\r\n\r\nããããããã§ã¯å·¥å€«ã§ããªããèããŠã¿ãŸãããïŒ\r\n$64,9,4$ ã¯ããããå¹³æ¹æ°ã§ãïŒãã®ããšã«æ°ã¥ãã°ïŒäœãšãªãå¹³æ¹æ ¹ããšã£ãŠã¿ãããªããŸãïŒæ®éã«æ£ã®å¹³æ¹æ ¹ãåããšïŒ\r\n$$g(3)=8, g(8)=3, g(13)=2$$\r\nãšãªã£ãŠïŒäœã®ããããã¿ããªãã®ã§ããïŒ$g(13)=2$ ã®ä»£ããã« $g(13)=-2$ ãšããŠãè¯ãããšã«æ°ã¥ãã°\r\n$$g(3)=8, g(8)=3, g(13)=-2$$\r\nãšãªãïŒ$g(x)=11-x$ ãäžã€ã®è§£ã ãšããããŸãïŒã€ãŸã $f(x)=(11-x)^2=(x-11)^2$ ãåŒ $(1)$ ã®äžã€ã®è§£ã§ããïŒãããçšããŠ\r\n$$P(x)=Q(x)(x-3)(x-8)(x-13)+(x-11)^2$$\r\nãšè¡šãããããšãããããŸãïŒ\r\n\r\n---\r\n\r\né¡é¡ãšããŠïŒ[176B](https:\\/\\/onlinemathcontest.com\\/contests\\/omc176\\/tasks\\/6981)ïŒ[B12D](https:\\/\\/onlinemathcontest.com\\/contests\\/omcb012\\/tasks\\/6768)ïŒ[190B](https:\\/\\/onlinemathcontest.com\\/contests\\/omc190\\/tasks\\/4774)ïŒ\\\r\nâ»190Bã¯æ¬åãšã¯å°ãéããããããŸãããïŒãããªãã«äŒŒãŠããã®ã§çœ®ããŸããïŒ",
"text": "(x-11)^2ã«ãã©ãçããŸã§",
"url": "https://onlinemathcontest.com/contests/omcb016/editorial/8193/566"
}
] | ã$x$ ã®æŽæ°ä¿æ°å€é
åŒ $P(x)$ ã
$$P(3)=64, \quad P(8)=9, \quad P(13)=4$$
ãã¿ãããšãïŒ$P(10)$ ã®ãšããã $1$ ä»¥äž $1000$ 以äžã®æŽæ°å€ã®ç·åãæ±ããŠäžããïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/6453 | C | OMCB016(C) | 200 | 96 | 147 | [
{
"content": "ãçŽç· $AB$ ãšçŽç· $CD$ ãšã®äº€ç¹ã $E$ ïŒçŽç· $AD$ ãšçŽç· $BC$ ãšã®äº€ç¹ã $F$ ãšããïŒ$\\cos{\\angle{BCD}}=\\dfrac{3}{5}$ ãã, $CE=5x,CF=5y$ ãšããã°ïŒ\r\n$$BC=3x,\\quad BE=4x,\\quad CD=3y,\\quad DF=4y$$\r\nãšãªãïŒæ¬¡ã«äžè§åœ¢ $ABF$ ãšäžè§åœ¢ $ADE$ ã¯çžäŒŒã§ããïŒæ¡ä»¶ããçžäŒŒæ¯ã¯ $9:7$ïŒãã£ãŠïŒ\r\n$$BF:DE=5y-3x:5x-3y=9:7$$\r\nã§ããããïŒããã解ãããšã§ $33x=31y$ ãåããïŒãããã£ãŠïŒä»¥äžãåŸãïŒ\r\n$$AB={\\dfrac{3}{4}}{BF}={\\dfrac{3}{4}}{(5y-3x)}=\\dfrac{54x}{31}$$\r\nåæ§ã«\r\n$$AD={\\dfrac{3}{4}}{DE}={\\dfrac{3}{4}}{(5x-3y)}=\\dfrac{42x}{31},\\quad CD=3y=\\dfrac{99x}{31}$$\r\nã§ããïŒãããã£ãŠïŒåè§åœ¢ $ABCD$ ã®é¢ç©ã«ã€ããŠä»¥äžã®åŒãæãç«ã€ïŒ\r\n$${\\dfrac{1}{2}}\\times{\\dfrac{54x}{31}}\\times3x+{\\dfrac{1}{2}}\\times{\\dfrac{42x}{31}}\\times{\\dfrac{99x}{31}}=4590$$\r\nããã解ããšïŒ$x=31$ ãåŸãããïŒ$AB=\\dfrac{54x}{31}=54, ~ BC=3x=93$ ãšãããïŒäžè§åœ¢ $ABC$ ã«ãããŠäžå¹³æ¹ã®å®çããæ±ããçã㯠$$AC^{2}=AB^{2}+BC^{2}=\\mathbf{11565}$$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb016/editorial/6453"
}
] | ãé¢ç©ã $4590$ ã§ããåè§åœ¢ $ABCD$ ã¯
$$\angle ABC = \angle ADC = 90^\circ,\quad AB:AD = 9:7,\quad \cos\angle BCD = \frac{3}{5}$$
ãæºãããŸãïŒãã®ãšãïŒç·å $AC$ ã®é·ãã®äºä¹ãæ±ããŠãã ããïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/3945 | D | OMCB016(D) | 200 | 156 | 253 | [
{
"content": "\r\n\r\nãäžå³ã®ããã«ãã¹ç®ã«ååãã€ããïŒ\\\r\nã$6$ ã®äœçœ®ãåºæºã«ããŠèããïŒãã以å€ã®æ°ãçŽ å æ°ã $2$ ã®çŽ¯ä¹ã® $\\\\{2,4,8\\\\}$ ã® $\\bigcirc$ ã°ã«ãŒãïŒçŽ å æ°ã $3$ ã®çŽ¯ä¹ã® $\\\\{3,9\\\\}$ ã® $\\times$ ã°ã«ãŒãïŒä»ã®æ°ãšå
±éã®çŽ å æ°ãæããªã $\\\\{1,5,7\\\\}$ ã® $\\bigtriangleup$ ã°ã«ãŒãã«åãããšïŒ\r\næ¥ããããšãã§ããªãã®ã¯ïŒ$6$ ãš $\\bigcirc$ïŒ$6$ ãš $\\times$ïŒ$\\bigcirc$ ã°ã«ãŒãå士ïŒ$\\times$ ã°ã«ãŒãå士ïŒãã£ãŠïŒ$6$ 㯠$E$ ã®ãã¹ã«ã¯æžãããšãã§ããªãïŒ\r\n- $6$ ã $A,C,G,I$ ã®ããããã«æžã蟌ãŸãããšããèãããïŒäŸãã° $A$ ã«ãããšãïŒä»ã®ã°ã«ãŒãã®æžã蟌ã¿æ¹ã¯äžã® $6$ éãïŒ\r\nããããã«ã€ã㊠$\\bigcirc$ ã°ã«ãŒãå
㧠$3!$ éãïŒ$\\times$ ã°ã«ãŒãå
㧠$2!$ éãïŒ$\\bigtriangleup$ ã°ã«ãŒãå
㧠$3!$ éãæžã蟌ãããšãã§ãïŒ$C,G,I$ ã« $6$ ããããã¿ãŒã³ãããã®ã§æžã蟌ã¿æ¹ã¯$$4\\times 6\\times 3!\\times 2!\\times 3!=1728$$éãïŒ\r\n\r\n- $6$ ã $B,D,F,H$ ã®ããããã«æžã蟌ãŸãããšããèãããïŒäŸãã° $B$ ã«ãããšãïŒä»ã®ã°ã«ãŒãã®æžã蟌ã¿æ¹ã¯äžã® $1$ éãïŒ\r\nããããã«ã€ã㊠$\\bigcirc$ ã°ã«ãŒãå
㧠$3!$ éãïŒ$\\times$ ã°ã«ãŒãå
㧠$2!$ éãïŒ$\\bigtriangleup$ ã°ã«ãŒãå
㧠$3!$ éãæžã蟌ãããšãã§ãïŒ$D,F,H$ ã« $6$ ããããã¿ãŒã³ãããã®ã§æžã蟌ã¿æ¹ã¯$$4\\times 3!\\times 2!\\times 3!=288$$éãïŒ\r\n\r\n\r\n以äžããæ±ããæžã蟌ã¿æ¹ã®ç·æ°ã¯ $1728+288=\\textbf{2016}$ éãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb016/editorial/3945"
}
] | ã $3\times 3$ ã®ãã¹ç®ãããïŒããããã®ãã¹ã« $1$ ä»¥äž $9$ 以äžã®æŽæ°ã $1$ åãã€æžã蟌ã¿ãŸãïŒã©ã®çžç°ãªããã¹ãçžç°ãªãæ°åãæžã蟌ãŸããŠãããšãïŒæ¬¡ã®æ¡ä»¶ãæºããæžã蟌ã¿æ¹ã¯å
šéšã§äœéããããŸããïŒ
- ç·åãå
±æãããã¹ã«æžããã $2$ åã®æŽæ°ã¯ãã¹ãŠäºãã«çŽ ïŒ
ãã ãïŒå転ãå転ã«ãã£ãŠäžèŽããæžã蟌ã¿æ¹ã¯åºå¥ããŸãïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/10374 | E | OMCB016(E) | 300 | 37 | 61 | [
{
"content": "ã$0 \\leq \\theta \\lt 2\\pi$ ãšããŠããïŒäžããããæ¹çšåŒãå€åœ¢ãããš\r\n$$(x-2)(x-(\\cos2\\theta+i \\sin2\\theta))(x-(\\cos2\\theta-i \\sin2\\theta)) = 0$$\r\nãšãªãããïŒäžè¬æ§ãã \r\n$$( \\alpha, \\beta, \\gamma ) = ( 2, \\cos2\\theta+i \\sin2\\theta, \\cos2\\theta -i \\sin2\\theta )$$ \r\nãšããŠããïŒãã®ãšãïŒ\r\n$$( \\alpha^n, \\beta^n, \\gamma^n ) = ( 2^n, \\cos2n\\theta+i \\sin2n\\theta, \\cos2n\\theta -i \\sin2n\\theta )$$\r\nãªã®ã§ïŒããããå
šãŠå®æ°ãšãªãããã®å¿
èŠååæ¡ä»¶ã¯ $2n\\theta = k\\pi$ ãšãªãæŽæ° $k$ ãååšããããšã ãšåããïŒãã®ãããªæå°ã® $n$ ã $100100$ ã§ããã®ã§ïŒããæŽæ° $l$ ã«ãã\r\n$$ 2\\theta = \\frac{l\\pi}{100100} $$\r\nãšãããïŒ$l$ ã $100100$ ãšäºãã«çŽ ã§ãªããš $n$ ã®æå°æ§ã«åããã®ã§ïŒ$l$ ãš $100100$ ã¯äºãã«çŽ ã§ããïŒéã«ãã®ãšã $n$ ã®æå°å€ã¯ããªãã $100100$ ã«ãªãïŒãã®ãšãïŒ\r\n$$\\alpha + \\beta + \\gamma = 2+2\\cos\\frac{l\\pi}{100100}$$ \r\nãšããŠããããå€ã¯ïŒ$0 \\leq l \\leq 100100$ ã〠$\\gcd(l,100100) = 1$ ãã¿ãã $l$ ã«å¯Ÿãããã®ãå
šãŠã§ããïŒãã®ãã㪠$l$ ã¯å
šéšã§ $\\phi(100100) = 28800$ åããïŒããããå°ããé ã« $l_1, l_2, \\ldots, l_{28800}$ ãšãããšïŒ\r\n$$\\cos\\frac{l_j\\pi}{100100}+\\cos\\frac{l_{28801-j}\\pi}{100100}=0 \\quad(\\because l_j+l_{28801-j}=100100)$$\r\nãæç«ããããšãåããããïŒ$2+2\\cos\\dfrac{l\\pi}{100100}$ ã®ç·å㯠$28800 \\times 2 = \\mathbf{57600}$ ãšãªãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb016/editorial/10374"
}
] | ã$\theta$ ãå®æ°ãšãïŒ$x$ ã«é¢ããæ¹çšåŒ
$$x^3 - (4 \cos^2Ξ) x^2 + (4 \cos{2Ξ} + 1)x - 2 = 0$$
ã®éè€ãå«ãã $3$ ã€ã®è€çŽ æ°è§£ã $α, β, γ$ ãšããŸãïŒãããšïŒ$α^n, β^n, γ^n$ ãããããå®æ°ãšãªããããªæ£æŽæ° $n$ ãååšãïŒãã®æå°å€ã¯ $100100$ ãšãªããŸããïŒãã®ãšãïŒ$α + β + γ$ ãšããŠããåŸãå€ã®ç·åã解çããŠäžããïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/6898 | F | OMCB016(F) | 400 | 15 | 26 | [
{
"content": "ãäžè§åœ¢ $FBQ$ ãšäžè§åœ¢ $FCQ$ ã®é¢ç©ã¯çããã®ã§ïŒä»¥äžãæç«ããïŒ\r\n$$\\frac12BF\\times BQ \\sin\\angle FBQ = \\frac12CF\\times CQ\\sin\\angle FCQ$$\r\nãŸãïŒ$\\sin\\angle FBQ = \\sin\\angle FCQ$ ã§ããããïŒä»¥äžãæç«ããïŒ\r\n$$BQ : CQ = CF : BF = AB : AC$$\r\nã§ããïŒããã«ïŒ$AB : AC = BD : CD$ ã§ããããïŒäžè§åœ¢ $ADQ$ ã®å€æ¥åã¯ïŒç·å $BC$ ã«å¯Ÿãã $AB : AC$ ã®ã¢ããããŠã¹ã®åã§ããïŒããã§ïŒç·å $BC$ ã«é¢ã㊠$A$ ãšå¯Ÿç§°ãªç¹ã $A^\\prime$ ãšãããšïŒåè§åœ¢ $BA^\\prime CF$ ã¯å¹³è¡å蟺圢ãªã®ã§ïŒ$A^\\prime$ ã¯çŽç· $FM$ äžã«ããïŒ$A^\\prime B : A^\\prime C = AB:AC$ ãªã®ã§ $A^\\prime$ ã¯äžè§åœ¢ $ADQ$ äžã«ããïŒåŸã£ãŠïŒ$R = A^\\prime$ ã§ããïŒä»ïŒäžç·å®çãã\r\n$$A^\\prime M = FM = AM = \\sqrt{\\frac12(AB^2 + AC^2) - \\bigg(\\frac12BC\\bigg)^2} = \\sqrt{\\frac{117}{2}}$$\r\nã§ããïŒãŸãïŒæ¹ã¹ãã®å®çãã\r\n$$QM = \\frac{BM\\times CM}{FM} = 16\\sqrt{\\frac{2}{117}}$$\r\nã§ããããïŒ\r\n$$QR^2 = \\bigg(\\sqrt{\\frac{117}{2}} - 16\\sqrt{\\frac{2}{117}}\\bigg)^2 = \\frac{7225}{234}$$\r\nã§ããïŒç¹ã«ïŒè§£çãã¹ãå€ã¯ $\\bf{7459}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb016/editorial/6898"
},
{
"content": "ãè§åºŠè¿œè·¡ã ãã§ã $MAïŒMR$ ã蚌æã§ããã®ã§ïŒçŽ¹ä»ããŠããïŒ \r\n\r\n---\r\n\r\nãçŽç· $AD$ ãšå $\\omega$ ã®äº€ç¹ã§ãã£ãŠïŒç¹ $A$ ã§ãªããã®ãç¹ $N$ ãšããïŒãã®ãšãïŒç¹ $N$ ã¯åŒ§ $BC$ ã®äžç¹ã«ããïŒ$BC \\perp MN$ ã§ããïŒ\r\n\r\nStep.1ã $4$ ç¹ $D,M,N,Q$ ã¯å
±åã§ããïŒ\\\r\nïŒèšŒæïŒçŽç· $MN$ ãšå $\\omega$ ã®äº€ç¹ã§ãã£ãŠïŒç¹ $N$ ã§ãªããã®ãç¹ $N^{\\prime}$ ãšããïŒ\\\r\nã$\\angle{FQN}+\\angle{ANN^{\\prime}}=\\angle{FQN}+\\angle{FNN^{\\prime}}=90^{\\circ}$ ã§ããïŒæåŸã®çå·ã¯ïŒåŒ§ $FN$ ã«å¯Ÿããååšè§ãšåŒ§ $FN^{\\prime}$ ã«å¯Ÿããååšè§ã®åã§ããããšããåŸãïŒïŒ\\\r\nãããã« $\\triangle DMN$ ãçŽè§äžè§åœ¢ã§ããããšãã $\\angle{MQN}=\\angle{MDN}$ ãåŸãïŒ\r\n\r\nStep.2ã$\\triangle ADM \\equiv \\triangle RDM$\\\r\nïŒèšŒæïŒçŽç· $AM$ ãšå $\\omega$ ã®äº€ç¹ã§ãã£ãŠïŒç¹ $A$ ã§ãªããã®ãç¹ $Q^{\\prime}$ ãšããïŒç¹ $Q$ ãšç¹ $Q^{\\prime}$ ã¯çŽç· $MN$ ã«å¯ŸããŠå¯Ÿç§°ãªäœçœ®ã«ååšãããã $\\angle{NAQ}=\\angle{NAQ^{\\prime}}$ïŒãã®ããšããïŒ$\\angle{DAM}=\\angle{DAQ}=\\angle{DRQ}=\\angle{DRM}$ ãåŸãïŒ\\\r\nããŸãïŒç·å¯Ÿç§°ã§ããããšãçšããã°ïŒ$\\angle{DMR}=\\angle{DMA}$ ãåŸãïŒ\\\r\nããã£ãŠïŒäžèŸºã®é·ããçããïŒ$DM$ å
±éïŒïŒå¯Ÿå¿ããäºã€ã®è§ãçããã®ã§ïŒ$\\triangle ADM \\equiv \\triangle RDM$ ã§ããïŒ\r\nã",
"text": "MAïŒMR ã®å¥èšŒæ",
"url": "https://onlinemathcontest.com/contests/omcb016/editorial/6898/564"
}
] | ã$BC = 8, CA = 10, AB = 7$ ã§ããäžè§åœ¢ $ABC$ ã®å€æ¥åã $\omega$ ãšããŸãïŒãŸãïŒ$\angle BAC$ ã®äºçåç·ãšèŸº $BC$ ã®äº€ç¹ã $D$ ãšãïŒ$\omega$ äžã®ç¹ $F$ ã $AF \parallel BC$ ãæºãããŠãããšããŸãïŒããã«ïŒèŸº $BC$ ã®äžç¹ã $M$ ãšãïŒ$FM$ ãš $\omega$ ã®äº€ç¹ã®ãã¡ $F$ ã§ãªãæ¹ã $Q$ïŒäžè§åœ¢ $ADQ$ ã®å€æ¥åãšçŽç· $FM$ ã®äº€ç¹ã®ãã¡ïŒ$Q$ ã§ãªãæ¹ã $R$ ãšããŸãïŒãã®ãšãïŒç·å $QR$ ã®é·ãã®äºä¹ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\cfrac{b}{a}$ ãšè¡šãããã®ã§ïŒ$a + b$ ã®å€ã解çããŠãã ããïŒ |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/7092 | A | OMC224(A) | 200 | 286 | 309 | [
{
"content": "**ééã£ã解æ³.**ãæ¿åºŠã®é«ããã®ããé ã«ç ç³æ°Ž $C, D, E$ ãéžãã§æ··ãåãããïŒãã®ãšãïŒåŸãããç ç³æ°Žã®æ¿åºŠã¯ïŒ\r\n$$\\dfrac{500 \\times 64 + 900 \\times 40 + 1000 \\times 36}{500 + 900 + 1000} = \\dfrac{130}{3} = 43.333...$$\r\nã§ããããïŒç¹ã«è§£çãã¹ãå€ã¯ $\\bf{4333}$ ã§ããïŒ\r\n\r\n**æ£ãã解æ³.**ãäžèšãééã£ã解æ³ãã§ã®çµã¿åããã§ã®èšç®ã«ããïŒæ±ããæ倧å€ã¯ $43\\\\%$ ããã倧ããããšããããïŒ \r\nã$45 \\\\%$ ã®ç ç³æ°Žãäœããšä»®å®ããŠïŒåç ç³æ°Žã§ç ç³ãäœ $\\rm{g}$ éäžè¶³ãããèãããšïŒç ç³æ°Ž $A$ 㧠$100 \\times (8-45) = -37\\rm{g}$ïŒç ç³æ°Ž $B$ 㧠$-63\\rm{g}$ïŒç ç³æ°Ž $C$ 㧠$+95\\rm{g}$ïŒç ç³æ°Ž $D$ 㧠$-45\\rm{g}$ïŒç ç³æ°Ž $E$ 㧠$-90\\rm{g}$ ãšãªãããšããããïŒ \r\nãåŸã£ãŠïŒç ç³æ°Ž $A, C, D$ ã®çµã¿åããã®ãšãã®ã¿ïŒæ¿åºŠã $45 \\\\%$ 以äžã«ãªãããšããããïŒãã®çµã¿åãããæ倧ã®å€ããšãïŒãã®äžã€ã®ç ç³æ°Žãçµã¿åãããŠäœã£ãæ°ããç ç³æ°Žã®æ¿åºŠã¯ä»¥äžã®ããã«èšç®ã§ããã®ã§ïŒè§£çãã¹ãå€ã¯ $\\bf{4587}$ ã§ããïŒ\r\n$$\\dfrac{100 \\times 8 + 500 \\times 64 + 900 \\times 40}{100 + 500 + 900} = \\dfrac{688}{15} = 45.866....$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/7092"
},
{
"content": "ãé«ã
${}\\_{5}\\mathrm{C}\\_{3}=10$ éããããªãã®ã§ïŒå
šãŠãè©Šãããšãå¯èœã§ããïŒ\\\r\nãããã $10$ éãå
šãŠãè©Šãã®ã¯ããé¢åãªã®ã§ïŒå·¥å€«ããããïŒæ¿åºŠãæ倧åããããã®å¿
èŠæ¡ä»¶ãèãããïŒ\r\n\r\nããŸãïŒæ¿åºŠãæ倧åããããã«ã¯ç ç³æ°Ž $C$ ãå
¥ããå¿
èŠãããïŒããã¯ä»¥äžã®ããã«èª¬æã§ããïŒ\\\r\nã$4$ ã€ã®ç ç³æ°Ž $X_1, X_2, X_3, X_4$ ãããïŒããããã®æ¿åºŠã $x_1, x_2 , x_3, x_4$ ãšããïŒãã®ãšãïŒ$X_1, X_2, X_3$ ãæ··ãåãããŠäœãããç ç³æ°ŽïŒ$X_2, X_3$ ãæ··ãåãããŠäœãããç ç³æ°ŽïŒ$X_2, X_3, X_4$ ãæ··ãåãããŠäœãããç ç³æ°Žã¯ïŒããšã«è¡ãã»ã©æ¿åºŠãé«ããªãïŒ$X_2, X_3$ ãæ··ããŠäœãããç ç³æ°Žãæ¹ã㊠$X_{2.5}$ ãªã©ãšçœ®ãçŽããšããããããïŒïŒãã£ãŠïŒç ç³æ°Ž $C$ ãå
¥ã£ãŠããªããã°ïŒ$3$ ã€ã®ç ç³æ°Žã®äžã§æ¿åºŠãæå°ã®ãã®ãé€ããŠïŒãã®ä»£ããã«ç ç³æ°Ž $C$ ãå«ããããšã§ïŒããæ¿åºŠãé«ãç ç³æ°Žãäœãããšãå¯èœã§ããïŒ\r\n\r\nã次ã«ïŒç ç³æ°Ž $D$ ã¯ç ç³æ°Ž $E$ ããåªå
ãããïŒããã¯ïŒ$D$ ãš $E$ ã«å
¥ã£ãŠããç ç³ã®éãçããïŒ$360 \\mathrm{g}$ïŒïŒæº¶æ¶²å
šäœã¯ $E$ ã®æ¹ãå€ãããšããåŸãïŒ\r\n\r\nã以äžã®ããšããïŒèããã¹ãæ··ãåããæ¹ã¯ïŒ$(A,B,C), (A,C, D), (B,C,D),(C,D,E)$ ã® $4$ éãã§ããïŒããšã¯å°éã«èšç®ããã°ããïŒ\r\n\r\nâ»ãã®ããã«å¿
èŠæ¡ä»¶ãèããŠã $6$ éãããæžããªããããšãã£ãŠãæåãã $10$ éãå
šãŠè©Šããæ¹ãè¯ãããšããã®ã¯ãããŸãããïŒ",
"text": "å°éã«è©Šãã®ã§ããã",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/7092/559"
}
] | ã倪éåã¯æ¿åºŠã $8\\%$ ã®ç ç³æ°Ž $A$ ã $100\rm{g}$ïŒ$24\\%$ ã®ç ç³æ°Ž $B$ ã $300\rm{g}$ïŒ$64\\%$ ã®ç ç³æ°Ž $C$ ã $500\rm{g}$ïŒ$40\\%$ ã®ç ç³æ°Ž $D$ ã $900\rm{g}$ïŒ$36\\%$ ã®ç ç³æ°Ž $E$ ã $1000\rm{g}$ ã® $5$ çš®é¡ã®ç ç³æ°ŽãçšæããŸããïŒ
ã倪éåã¯ãã®äžã® $3$ çš®é¡ã®ç ç³æ°Žãéžã³ïŒãã®å
šãŠãæ··ãåãããŠæ°ããç ç³æ°Žãäœãããšã«ããŸããïŒæ°ããç ç³æ°Žã®æ¿åºŠãšããŠèãããããã®ã®å
æ倧ã®ãã®ã¯ $x [\\%]$ ã§ãããšããŸãïŒ$100x$ ã $10^{-1}$ ã®äœã§åæšäºå
¥ããå€ã解çããŠãã ããïŒ |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/8687 | B | OMC224(B) | 200 | 261 | 286 | [
{
"content": "ã$N=2$ ã®ãšãã¯æããã« $f(N) = 1$ ã§ããïŒä»¥äžïŒ$N\\geq 3$ ãšããïŒ\\\r\nãå¥æ°åã®æäœã®åŸã§ $A$ 㯠$(N,N-1,\\ldots,1)$ ãå·¡åããããã®ïŒå¶æ°åã®æäœã®åŸã§ $A$ 㯠$(1,2,\\ldots,N)$ ãå·¡åããããã®ã«ãªãããïŒ$f(N)$ ã¯å¶æ°ã§ããïŒããŸïŒé£ç¶ãã $2$ åã®æäœã¯ãæ«å°Ÿã® $2$ é
ãé çªãå€ããã«å
é ã«ç§»ãããšãŸãšããããïŒãã®è¡šçŸã«ããïŒ$f(N)=\\mathrm{lcm}(N,2)$ ã§ããããšããããïŒ\\\r\nã$N$ ã®å¶å¥ã§å ŽååãããŠç·åãæ±ããããšã§ïŒæ±ããå€ã¯ $\\textbf{7547}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/8687"
}
] | ãé·ã $N$ ã®æ°å $A$ ãããïŒã¯ãã㯠$(1,2,\dots,N)$ ã§ãïŒ$A$ ã«å¯Ÿãã以äžã®æäœãïŒã¯ãã㊠$(1,2,\dots,N)$ ã«åã³æ»ããŸã§ç¹°ãè¿ãè¡ããŸãïŒ
- å¥æ°åç®ã®æäœã§ã¯ïŒåãã $N-1$ é
ã®äžŠã³ãååŸéã«ããïŒ
- å¶æ°åç®ã®æäœã§ã¯ïŒåŸããã $N-1$ é
ã®äžŠã³ãååŸéã«ããïŒ
è¡ãããæäœã®åæ°ã $f(N)$ ãšãããšãïŒ$f(2)+f(3)+\cdots+f(99)+f(100)$ ãæ±ããŠãã ããïŒ |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/8724 | C | OMC224(C) | 300 | 134 | 203 | [
{
"content": "ãçµè«ããè¿°ã¹ããšïŒå
æ倪éåãåãŠãããšã¯ä»¥äžãšåå€ã§ããïŒä»¥äžïŒããã**åå©æ¡ä»¶**ãšãã¶ïŒ\r\n\r\n- $A$ ã®ã¡ããã© $1$ é
ã®ã¿ã $\\bmod\\ 3$ 㧠$2$ ã§ããïŒæ®ãã¯ãã¹ãŠ $\\bmod\\ 3$ 㧠$0$ ã§ããïŒ\r\n\r\nããŸãïŒåå©æ¡ä»¶ãæºããããŠãããšãïŒå
æ倪éåã¯ãã¹ãŠã®é
ã$\\bmod\\ 3$ 㧠$0$ ã§ãããããªç¶æ
ã«ã§ããïŒãã®åŸã¯ïŒçŽåã«åŸæ次éåãéžãã é
ãéžã³ç¶ããã°ããïŒ\\\r\nãããããµãŸãããšïŒåŸæ次éå㯠$\\bmod\\ 3$ 㧠$0$ ã§ãªãé
ãå«ãŸããç¶æ
ã§æäœãç¶ããå¿
èŠãããïŒåå©æ¡ä»¶ãæºããããŠããªããšãã«ã¯ïŒãã®çŽåŸã®å
æ倪éåã®äžåã®æäœã«ããããããæºããããããšã«æ³šæããã°ïŒåŸæ次éåã¯ä»¥äžã®ããã«æäœããã°åã€ããšãã§ããïŒ\r\n\r\n- $\\bmod\\ 3$ 㧠$2$ ã®é
ãäžã€ä»¥äžããå Žåã¯ïŒãã®ãã¡äžã€ãä»»æã«éžã¶ïŒ\r\n- $\\bmod\\ 3$ 㧠$1$ ã®é
ãäžã€ä»¥äžããå Žåã¯ïŒãã®ãã¡äžã€ãä»»æã«éžã³ïŒïŒããã«å¯ŸããŠæäœããã®ã§ã¯ãªãïŒããã«å¯ŸããŠã¯æäœããªãããã«ããïŒ\r\n\r\nã以äžã«ããïŒæ±ããå Žåã®æ°ã¯ $11 \\times 4\\times 3^{10} = \\textbf{2598156}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/8724"
}
] | ãé·ã $11$ ã®æŽæ°å $A=(A_1,A_2,\dots,A_{11})$ ãçšããŠïŒå
æ倪éåãšåŸæ次éåãã²ãŒã ãããŸãïŒå
æ倪éåããå§ããŠïŒããããã«èš±ããã以äžã®æäœã亀äºã«è¡ãïŒå
ã«æäœãã§ããªããªã£ãæ¹ãè² ãã§ãïŒ
- å
æ倪éåïŒ$1 \le i \le 11$ ã〠$A_i \ge 2$ ãªãæŽæ° $i$ ãä»»æã«äžã€éžã³ïŒ$A_i$ ã $2$ æžããïŒ
- åŸæ次éåïŒ$1 \le i \le 11$ ã〠$A_i \ge 1$ ãªãæŽæ° $i$ ãä»»æã«äžã€éžã³ïŒ$A_i$ ã $1$ æžããïŒ
$A_1,A_2,\ldots,A_{11}$ ããã¹ãŠ $1$ ä»¥äž $11$ 以äžã§ãããã㪠$A$ 㯠$11^{11}$ åãããŸããïŒãã®ãã¡åŸæ次éåã®æäœã«ãããå
æ倪éåãåãŠããã®ã¯ããã€ãããŸããïŒ |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/6710 | D | OMC224(D) | 400 | 12 | 31 | [
{
"content": "<details>\r\n<summary>ã·ã ãœã³ç·ã«ã€ããŠ<\\/summary>\r\näžè¬ã«ïŒäžè§åœ¢ $XYZ$ ã®å€æ¥åäžã®ç¹ $W$ ããäžè§åœ¢ $XYZ$ ã®å蟺 (ã延é·ããçŽç·) ã«äžãããåç·ã®è¶³ãã¡ã¯ïŒåäžçŽç·äžã«ããïŒãã®çŽç·ã®ããšãïŒäžè§åœ¢ $XYZ$ ã«å¯Ÿãã $W$ ã®ã·ã ãœã³ç·ãšããïŒæ¬åã«ãããŠã¯ïŒ$3$ ç¹ $P, Q, R$ ã¯äžè§åœ¢ $ABC$ ã«å¯Ÿãã $D$ ã®ã·ã ãœã³ç·ããªãïŒ\r\n<\\/details>\r\n\r\nã$4$ ç¹ã®çµ $(B,D,P,R), (C,D,P,Q)$ ã¯ïŒããããç·å $BD, CD$ ãçŽåŸãšããåäžã«ååšããïŒåŸã£ãŠïŒæ£åŒŠå®çããïŒ\r\n$$\\sin\\angle PDR = \\frac{PR}{BD} = \\frac{4}{5},\\quad \\sin\\angle PDQ = \\frac{PQ}{CD} = \\frac{20}{29}$$\r\nã§ããïŒãŸãïŒ$\\angle PDR = \\angle ABC \\lt 90^\\circ, \\angle PDQ = \\angle ACB\\lt90^\\circ$ ããïŒ\r\n$$\\cos\\angle PDR = \\frac{3}{5},\\quad \\cos\\angle PDQ = \\frac{21}{29}$$\r\nã§ããïŒããã§ïŒ\r\n$$\\angle BDC = \\angle ADB + \\angle ADC = \\angle ACB + \\angle ABC = \\angle PDQ + \\angle PDR$$\r\nã§ããããïŒäœåŒŠå®çããïŒ\r\n$$BC^2 = BD^2 + CD^2 - 2BD\\times CD\\cos(\\angle PDQ + \\angle PDR) = 1636$$\r\nã§ããïŒãŸãïŒ$3$ ç¹ $P,Q,R$ ã¯åäžçŽç·äžã«ããã®ã§ïŒ \r\n$$\\angle DQP = \\angle DCP = \\angle DAB,\\quad \\angle DPQ = 180^\\circ - \\angle DPR = 180^\\circ - \\angle DBR = \\angle DBA$$\r\nããïŒäžè§åœ¢ $ABD$ ãšäžè§åœ¢ $QPD$ ã¯çžäŒŒã§ããïŒåæ§ã«ïŒäžè§åœ¢ $ACD$ ãšäžè§åœ¢ $RPD$ ã¯çžäŒŒãªã®ã§ïŒ\r\n$$AB : AC = \\frac{PQ\\times BD}{DP} : \\frac{PR\\times CD}{DP} = BD : CD = 25 : 29$$\r\nã§ããïŒåŸã£ãŠïŒæ±ããçãã¯ïŒ\r\n$$\\bigg(\\frac{25 - 29}{25+29}BC\\bigg)^2 = \\frac{6544}{729}$$\r\nã§ããïŒç¹ã«ïŒè§£çãã¹ãå€ã¯ $\\bf{7273}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/6710"
},
{
"content": "ã$\\sin{\\angle{ABC}}=\\sin{\\angle{PDR}}=\\dfrac{20}{25},\\sin{\\angle{ACB}}=\\sin{\\angle{PDQ}}=\\dfrac{20}{29}$ ããïŒ$A$ ãã $BC$ ã«äžãããåç·ã®è¶³ã $H$ ãšãããšïŒ$AB=25k,AC=29k,AH=20k$ ãšããïŒ$BH=15k,CH=21k$ ãšãªãïŒ \r\nããããäœåŒŠå®çãçšã㊠$\\cos{\\angle{BAC}}$ ãæ±ããããšã§ $\\cos{\\angle{BDC}}(=-\\cos{\\angle{BAC}})$ ã®å€ããããïŒäžè§åœ¢$BCD$ ã§äœåŒŠå®çãã $BC$ ã®é·ããæ±ãããïŒããšã¯å
¬åŒè§£èª¬ãšåæ§ã«ããŠçããå°ãããšãåºæ¥ãïŒ",
"text": "äžè§åœ¢ ABC ã®äžèŸºæ¯ã®æ±ºå®ã®ä»æ¹ã®å¥è§£",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/6710/567"
}
] | ãéè§äžè§åœ¢ $ABC$ ã®å€æ¥åã® $A$ ãå«ãŸãªã匧 $BC$ äžã«ç¹ $D$ ãåãïŒ$D$ ããçŽç· $BC, CA, AB$ ã«äžãããåç·ã®è¶³ããããã $P,Q,R$ ãšãããšïŒä»¥äžãæç«ããŸããïŒ
$$BD = 25,\quad CD = 29,\quad PQ = PR = 20$$
ã$\angle BAC$ ã®äºçåç·ãšèŸº $BC$ ãšã®äº€ç¹ã $E$ ãšãããšãïŒ$(BE - CE)^2$ ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/7255 | E | OMC224(E) | 400 | 100 | 202 | [
{
"content": "ã$2$ 以äžã®æ£æŽæ° $n$ ã«å¯ŸãïŒ$g(n)$ ã以äžã®ããã«å®ããïŒ\r\n- $n$ ãçŽ æ°ãªãã°ïŒ$g(n) = -n$ ãšããïŒ\r\n- $n$ ãçŽ æ°ã§ãªããªãã°ïŒ$g(n)$ 㯠$n$ ã®æ£ã®çŽæ°ã®ãã¡ $3$ çªç®ã«å°ããæ°ãšããïŒ\r\n\r\nãã®ãšã $f(n)g(n)=n$ ãæãç«ã€ïŒããŸïŒ$g(k)$ ã $3$ ã®åæ°ãšãªãã®ã¯æ¬¡ã® $3$ éãã®ç¶æ³ã«éãããïŒ\r\n\r\n- $k=3$ ã®ãšã $g(k)=-3$ ãšãªãïŒ\r\n- $6\\mid k$ ã®ãšã $g(k)=3$ ãšãªãïŒ\r\n- $2\\nmid k$ ã〠$5\\nmid k$ ã〠$7\\nmid k$ ã〠$9\\mid k$ ã®ãšã $g(k)=9$ ãšãªãïŒ\r\n\r\nç¹ã« $g(k)=9$ ãªã $k$ ã¯ïŒäžåœå°äœå®çãã $630$ ãæ³ãšã㊠$1\\times 1\\times 4\\times 6=24$ åååšããïŒãŸãïŒ$631$ ä»¥äž $800$ 以äžã®æŽæ°ã§ãã£ãŠ $3$ ã€ç®ã®æ¡ä»¶ãæºãããã®ã®æ°ã¯ïŒ$1$ ä»¥äž $170$ 以äžã® $9$ ã®åæ°ã§ãã£ãŠ $2, 5, 7$ ã®ãããã§ãå²ãåããªããã®ã®æ°ãšçããïŒãã㯠$9, 27, 81, 99, 117, 153$ ã® $6$ ã€ã®ã¿ã§ããïŒãã£ãŠïŒ$n$ ã $3$ ã§å²ãåããæ倧ã®åæ°ã $v_3(n)$ ãšãããšïŒ\r\n$$v_3 \\left\\(\\prod_{k=2}^{800}g(k)\\right\\)\r\n=1+\\bigg\\lfloor\\frac{800}{6}\\bigg\\rfloor+2(24 + 6) = 194\r\n$$\r\n\r\nã§ããïŒä»¥äžããïŒæ±ããçãã¯\r\n$$\\begin{aligned}\r\nv_3\\left\\(\\prod_{k=2}^{800}f(k)\\right\\)\r\n= v_3\\left\\(\\prod_{k=2}^{800}\\frac{k}{g(k)}\\right\\)\r\n =v_3(800!) - 194 = \\bf202\r\n\\end{aligned}$$\r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/7255"
}
] | ã$2$ 以äžã®æŽæ° $n$ ã«å¯ŸãïŒ$n$ ãå²ãåãæŽæ°ã®ãã¡ $3$ çªç®ã«å€§ãããã®ã $f(n)$ ãšããŸãïŒäŸãã°ïŒ$f(2)=-1, f(22)=2, f(224)=56$ ã§ãïŒãã®ãšãïŒ
$$\prod_{k=2}^{800}f(k)$$
ã $3$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠãã ããïŒ |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/9795 | F | OMC224(F) | 500 | 10 | 26 | [
{
"content": "ãäžåŒã§ $n = 1$ ã®å ŽåãèãïŒ$a_1 = 0$ ã§ããïŒãŸãïŒ$m$ ã $2$ ä»¥äž $N$ 以äžã®æŽæ°ãšãããšãïŒäžåŒã§ $n = m$ ã®å Žåãã $n = m-1$ ã®å ŽåãåŒãããšã§ïŒ\r\n$$\\sum_{d\\mid m} a_d = \\begin{cases}\r\n0 &(m \\equiv 1 \\mod 3)\\\\\\\\\r\n1 &(m \\not\\equiv 1 \\mod 3)\r\n\\end{cases}\\tag1$$\r\nãåŸãïŒ$a_1 = 0$ ãš $(1)$ ãåæã«æºãããã㪠$a_1, a_2, \\ldots, a_{N}$ ã¯äžæã«å®ãŸãããšã«æ°ãã€ããïŒ\\\r\nãããã§ïŒ$n$ ãä»»æã® $1$ ä»¥äž $N$ 以äžã®æŽæ°ãšãïŒ$n = p_1^{e_1}p_2^{e_2}\\cdots p_t^{e_t}$ ãçŽ å æ°å解ãšãããšãïŒ$b_n$ ã次ã®ããã«å®ããïŒ\r\n- $n = 1$ ã®ãšã $b_n = 0$ ãšãïŒ$n = 3$ ã®ãšã $b_n = 1$ ãšããïŒ\r\n- $n$ ã $9$ ã®åæ°ã§ãããïŒ$p_1, p_2,\\ldots, p_t$ ã®äžã« $3$ ã§å²ã£ãŠ $1$ äœãçŽ æ°ãå«ãŸãããšãïŒ$b_n = 0$ ãšããïŒ\r\n- $n$ ã $3$ ã®åæ°ã§ãªãïŒ$p_1, p_2,\\ldots, p_t$ ã®äžã« $3$ ã§å²ã£ãŠ $1$ äœãçŽ æ°ãå«ãŸããªããšãïŒ$b_n = (-1)^{e_1 + e_2 + \\cdots + e_t + 1}2^{t - 1}$ ãšããïŒ\r\n- $n$ ã $3$ ã®åæ°ã§ããã $9$ ã®åæ°ã§ãªãïŒ$p_1, p_2,\\ldots, p_t$ ã®äžã« $3$ ã§å²ã£ãŠ $1$ äœãçŽ æ°ãå«ãŸããïŒ$t \\ge 2$ ã§ãããšãïŒ$b_n = (-1)^{e_1 + e_2 + \\cdots + e_t + 1}2^{t - 2}$ ãšããïŒ\r\n\r\nã以äžïŒãã®ããã«å®ãã $b_n$ ãã¡ã $a_n$ ãã¡ã®æ¡ä»¶ïŒã€ãŸãïŒ$b_1 = 0$ ãš $(1)$ ãåæã«æºããããšã瀺ãïŒ\\\r\nããŸãïŒ$a_1 = b_1 = 0$ ã§ããïŒ$(1)$ 㧠$m = 3$ ã®å Žåãèããã°ïŒ$a_3 = b_3 = 1$ ãšãªãããšããããïŒ\\\r\nã$n$ ã $9$ ã®åæ°ã®å ŽåïŒ\r\n$$\\sum_{d\\mid n} b_d\r\n= \\sum_{d\\mid (n\\/3^{v_3(n) - 1})} b_d + \\sum_{9\\mid d\\mid n} b_d\r\n= \\sum_{d\\mid (n\\/3^{v_3(n) - 1})} b_d$$\r\nãšãªãïŒ$n$ ã $3$ ã®åæ°ã§ããã $9$ ã®åæ°ã§ãªãå Žåã«åž°çãããïŒ\\\r\nã$n$ ã $3$ ã§å²ã£ãŠ $1$ äœãçŽ å æ°ãæã£ãŠãããšãïŒãã®ãã¡äžã€ã $p$ ãšãããšïŒ\r\n$$\\sum_{d\\mid n} b_d\r\n= \\sum_{d\\mid (n\\/p^{v_p(n)})} b_d + \\sum_{p\\mid d\\mid n} b_d\r\n= \\sum_{d\\mid (n\\/p^{v_p(n)})} b_d$$\r\nãšãªãïŒ$n$ ã $3$ ã§å²ã£ãŠ $1$ äœãçŽ å æ°ãæããªãå Žåã«åž°çãããïŒ\\\r\nã$n$ ã $3$ ã®åæ°ã§ããã $9$ ã®åæ°ã§ãªãïŒ$3$ ã§å²ã£ãŠ $1$ äœãçŽ å æ°ãæããªããšãïŒ\r\n$$\\sum_{d\\mid n} b_d\r\n= \\sum_{d\\mid (n\\/3)} (b_d + b_{3d})\r\n= b_1 + b_3 + \\sum_{1 \\lt d\\mid (n\\/3)} (b_d - b_d)\r\n= 1$$\r\nãšãªãïŒæ¡ä»¶ãæºããïŒ\\\r\nã$n$ ã $3$ ã®åæ°ã§ãªãïŒ$3$ ã§å²ã£ãŠ $1$ äœãçŽ å æ°ãæããªããšãïŒ$n$ ã®çŽ å æ°ã®ãã¡ $1$ ã€ã $p$ ãšãããšïŒ\r\n$$\\begin{aligned}\r\n\\sum_{d\\mid n} b_d\r\n&= \\sum_{d\\mid (n\\/p^{v_p(n)})} \\sum_{k = 0}^{v_p(n)}b_{p^kd}\\\\\\\\\r\n&= \\sum_{k = 1}^{v_p(n)}b_{p^k}+\r\n\\sum_{1 \\lt d\\mid (n\\/p^{v_p(n)})} \\Bigg(b_d + \\sum_{k = 1}^{v_p(n)}(-1)^k2b_{d}\\Bigg)\\\\\\\\\r\n&= \\begin{cases}\r\n\\displaystyle\\sum_{d\\mid (n\\/p^{v_p(n)})} b_d &(v_p(n) \\equiv 0 \\mod 2)\\\\\\\\\r\n\\displaystyle1 - \\sum_{d\\mid (n\\/p^{v_p(n)})} b_d &(v_p(n) \\equiv 1 \\mod 2)\r\n\\end{cases}\r\n\\end{aligned}$$\r\nãšãªãïŒããã§ïŒ$v_p(n)$ ãå¶æ°ãªãã° $n$ ãš $n\\/p^{v_p(n)}$ ã $3$ ã§å²ã£ãäœãã¯çããïŒå¥æ°ãªãã°çãããªãããšã«æ°ãã€ãããšïŒ$b_1 = 0$ ãšããããŠæ¡ä»¶ãæºããããšããããïŒ\\\r\nã以äžããïŒãã® $b_n$ ãã¡ãåé¡ã® $a_n$ ãã¡ã§ããããšãããã£ãïŒããŠïŒ$3$ ã§å²ã£ãŠ $2$ äœãçŽ æ°ã¯å°ããé ã« $2, 5, 11, 17, 23, 29,41,47,...$ ã§ãããïŒ\r\n$$2\\times 5\\times 11\\times\\cdots \\times41 \\lt N \\lt 2^2\\times 5\\times 11 \\times\\cdots \\times 41$$\r\nã§ããã®ã§ïŒ$a_N$ ãã¡ã®äžã«çŸããå®æ°ã¯ $0, \\pm1, \\pm 2, \\pm 4, \\ldots, \\pm32, 64$ ã®ã¿ã§ããïŒãã£ãŠïŒæ±ããç·å㯠$\\bf190$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/9795"
},
{
"content": "$D_3(n)$ ãïŒ$n$ ã3ã®åæ°ã®ãšã1ïŒããã§ãªããšã0ã§ããé¢æ°ãšããŸãïŒä»¥é $n \\gt 1$ ãšããŸãïŒ \\\r\n$\\sum_{d|n} a_d = 1 - D_3(n+2)$ ãšãªãïŒã¡ããŠã¹ã®å転å
¬åŒãã $$a_n = \\sum_{d|n} \\mu(n\\/d) (1-D_3(d+2)) = - \\sum_{d|n} \\mu(n\\/d) D_3(d+2)$$ ãšãªããŸã( $\\mu(n)$ ã¯ã¡ããŠã¹é¢æ°)ïŒ\\\r\nããã§ïŒ$D_3(n+2)$ ã¯ã»ãšãã©ä¹æ³çãªé¢æ°ã§ãïŒã€ãŸãïŒ$i,j$ ãã©ã¡ãã3ã§å²ã£ãŠ2äœãå Žåãé€ã $D_3(i+2)D_3(j+2) = D_3(ij+2)$ ã§ãïŒããããïŒ$i,j$ ãäºãã«çŽ ãªãïŒ$i,j$ ã®ã©ã¡ãã3ã§å²ã£ãŠ2äœãçŽ æ°ãçŽ å æ°ã«æã€å Žåãé€ã㊠$(-a_i)(-a_j) = -a_{ij}$ ã§ããããšãããããŸãïŒãã£ãŠ $a_n$ ãæ±ããããã«ã¯\r\n- $n$ ã3ã§å²ã£ãŠ1äœãçŽ æ°ã®åªä¹ã§ããå Žå\r\n- $n$ ã3ã®åªä¹ã§ããå Žå\r\n- $n$ ã3ã§å²ã£ãŠ2äœãçŽ æ°ã®ç©ã®å Žå\r\n\r\nã® $a_n$ ãåããã°è¯ãã§ãïŒåäºã€ã¯ã¡ããŠã¹é¢æ°ã®å®çŸ©ãç¥ã£ãŠããã°æ±ãŸãã®ã§ $n$ ã3ã§å²ã£ãŠ2äœãçŽ æ°ã®ç©ã®å Žåã ãæ±ããŸãïŒ\\\r\n$n = \\prod_{i = 1}^m p_i^{e_i} \\hspace{8pt} (e_i\\gt 0,p_iã¯çžç°ãªã3ã§å²ã£ãŠ2äœãçŽ æ°)$ ã®ãšã $n_0 = \\prod_{i = 1}^m p_i$ ãšãããšïŒ\r\n$$a_n = - \\sum_{d|n} \\mu(n\\/d) D_3(d+2) = - \\sum_{d|n} \\mu(d) D_3(n\\/d+2) = - \\sum_{d|n_0} \\mu(d) D_3(n\\/d+2)$$\r\nããã§ïŒ$\\omega (n)$ ã $n$ ãå²ãåãçŽ æ°ã®åæ°ïŒ$\\Omega (n)$ ã $n$ ã®çŽ å æ°ã®éè€ãå«ããåæ°ãšãããšïŒåŒäžã® $\\mu(d)$ 㯠$(-1)^{\\omega (d)}$ ã«çããïŒ$D_3(n\\/d+2)$ 㯠$((-1)^{\\Omega(n\\/d)} + 1)\\/2$ ã«çããã§ãïŒãã£ãŠ\r\n$$\r\n\\begin{aligned}\r\na_n = - \\sum_{d|n_0} (-1)^{\\omega (d)} ((-1)^{\\Omega(n\\/d)} + 1)\\/2 &= - \\frac{1}{2} \\sum_{d|n_0} (-1)^{\\omega (d)} (-1)^{\\Omega(n\\/d)} &- \\frac{1}{2} \\sum_{d|n_0} (-1)^{\\omega (d)} \\\\\\\\\r\n&= - \\frac{(-1)^{\\Omega(n)}}{2} \\sum_{d|n_0} 1 - 0 \\\\\\\\\r\n&= (-1)^{\\Omega(n) + 1} 2^{\\omega(n) - 1}\r\n\\end{aligned}\r\n$$",
"text": "äžè¬é
ã®èŠã€ãæ¹",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/9795/561"
},
{
"content": "ã$n\\geq 2$ ã«ã€ããŠèããïŒ$n = p_1^{e_1}p_2^{e_2}\\cdots p_t^{e_t}$ ãšãããšïŒå
é€åçã®èŠé ã§\r\n$$\r\na_n\r\n=\\displaystyle\\sum_{d|n}a_d-\\sum_{1\\leq i\\leq t}\\sum_{d|\\frac{n}{p_i}}a_d+\\sum_{1\\leq i\\lt j\\leq t}\\sum_{d|\\frac{n}{p_ip_j}}a_d-\\cdots +(-1)^t\\sum_{d|\\frac{n}{p_1p_2\\cdots p_t}}a_d\r\n$$\r\nããšå€åœ¢ã§ããïŒ\r\n<details> <summary> å
·äœäŸ <\\/summary>\r\n\r\nã$n=pq$ ã®ãšã\r\n$$\r\na_{pq}\r\n=(a_{pq}+a_p+a_q+a_1)-(a_p+a_1)-(a_q+a_1)+a_1$$\r\n\r\nã$n=p^2q$ ã®ãšã\r\n$$\r\na_{p^2q}=(a_{p^2q}+a_{p^2}+a_{pq}+a_p+a_q+a_1)-(a_{pq}+a_p+a_q+a_1)-(a_{p^2}+a_p+a_1)+(a_p+a_1)$$\r\n<\\/details>\r\n\r\n$$\r\n\\displaystyle\\sum_{d|m}a_d\r\n=\r\n\\begin{cases}\r\n0 & (m \\equiv 1 \\mod 3)\\\\\\\\\r\n1 & (m \\not\\equiv 1 \\mod 3)\r\n\\end{cases}$$\r\n\r\nãã¯æ±ãã¥ããã®ã§ïŒäž¡èŸºã $2$ åããŠå³èŸºãã $(1-1)^t$ ãåŒããš\r\n\r\n$$\r\n2a_n\r\n=S_n-\\sum_{1\\leq i\\leq t}S_{\\frac{n}{p_i}}+\\sum_{1\\leq i\\lt j\\leq t}S_{\\frac{n}{p_ip_j}}-\\cdots +(-1)^tS_{\\frac{n}{p_1p_2\\cdots p_t}}$$\r\n\r\nããã ãïŒ\r\n\r\n$$\r\nS_m=\r\n2\\sum_{d|m}a_d-1=\r\n\\begin{cases}\r\n-1 & (m \\equiv 1 \\mod 3)\\\\\\\\\r\n1 & (m \\not\\equiv 1 \\mod 3)\r\n\\end{cases}$$\r\n\r\nããšããïŒ\r\n- $n$ ã $3$ ã®åæ°ã§ãªããšã\r\n\r\nã$i,j$ ããšãã« $3$ ã®åæ°ã§ãªããšãïŒ$(-S_i)(-S_j)=-S_{ij}$ \\\r\nããŸããããã $S_{\\frac{n}{m}}S_m=-S_n$ ïŒãã£ãŠ $S_{\\frac{n}{m}}=-S_nS_m$ ãæãç«ã€ããšã«æ³šæããã°\r\n\r\n$$\\begin{aligned}\r\n2a_n\r\n&=S_n-\\sum_{1\\leq i\\leq t}S_{\\frac{n}{p_i}}+\\sum_{1\\leq i\\lt j\\leq t}S_{\\frac{n}{p_ip_j}}-\\cdots +(-1)^tS_{\\frac{n}{p_1p_2\\cdots p_t}}\\\\\\\\\r\n&=-S_n\\left\\lbrace S_1-\\sum_{1\\leq i\\leq t}S_{p_i}+\\sum_{1\\leq i\\lt j\\leq t}S_{p_ip_j}-\\cdots +(-1)^tS_{p_1p_2\\cdots p_t}\\right\\rbrace\\\\\\\\\r\n&=S_n(1+S_{p_1})(1+S_{p_2})\\cdots (1+S_{p_t})\r\n\\end{aligned}$$\r\nããã®å€ã¯\\\r\nãã$p_i\\equiv 1(\\text{mod} 3)$ ãªã $n$ ã®çŽ å æ° $p_i$ ãååšãããšã $0$\\\r\nããããã§ãªããšã $a_n=(-1)^{e_1+e_2+\\cdots +e_t+1}2^{t-1}$ ãšãªãïŒ\r\n\r\n- $n$ ã $3$ ã®åæ°ã§ãã£ãŠ $9$ ã®åæ°ã§ãªããšã\r\n\r\nã$p_1=3$ ãšãããšïŒ\r\n$$\\begin{aligned}\r\n2a_n\r\n&=S_n-\\sum_{1\\leq i\\leq t}S_{\\frac{n}{p_i}}+\\sum_{1\\leq i\\lt j\\leq t}S_{\\frac{n}{p_ip_j}}-\\cdots +(-1)^tS_{\\frac{n}{p_1p_2\\cdots p_t}}\\\\\\\\\r\n&=\\left\\lbrace S_n-\\sum_{1\\lt i\\leq t}S_{\\frac{n}{p_i}}+\\sum_{1\\lt i\\lt j\\leq t}S_{\\frac{n}{p_ip_j}}-\\cdots (-1)^{t-1}S_{\\frac{n}{p_2p_3\\cdots p_t}}\\right\\rbrace - \\left\\lbrace S_{\\frac{n}{p_1}}-\\sum_{1=i\\lt j\\leq t}S_{\\frac{n}{p_ip_j}}+\\cdots +(-1)^{t-1}S_{\\frac{n}{p_1p_2\\cdots p_t}}\\right\\rbrace\r\n\\end{aligned}$$\r\n\r\nã$2$ è¡ç®ã®ååã®é
ã¯ãã¹ãŠ $3$ ã®åæ°ãªã®ã§ $(1-1)^{t-1}$ ãšçããïŒåŸåã®é
㯠$2a_{\\frac{n}{3}}$ ãšçããããïŒ\r\n$$a_n=-a_{\\frac{n}{3}}$$\r\nããã ã $a_3=1$ ã§ããïŒ\r\n\r\n- $n$ ã $9$ ã®åæ°ã®ãšã\r\n\r\n$$\\begin{aligned}\r\n2a_n\r\n&=S_n-\\sum_{1\\leq i\\leq t}S_{\\frac{n}{p_i}}+\\sum_{1\\leq i\\lt j\\leq t}S_{\\frac{n}{p_ip_j}}-\\cdots +(-1)^tS_{\\frac{n}{p_1p_2\\cdots p_t}}\\\\\\\\\r\n&=1-\\sum_{1\\leq i\\leq t}1+\\sum_{1\\leq i\\lt j\\leq t}1-\\cdots +(-1)^t\\\\\\\\\r\n&=(1-1)^t=0\r\n\\end{aligned}$$",
"text": "äžè¬é
ã®èŠã€ãæ¹(å¥è§£)",
"url": "https://onlinemathcontest.com/contests/omc224/editorial/9795/562"
}
] | ã$N = 10^{8}$ ãšããŸãïŒ$N$ åã®å®æ° $a_1, a_2, \ldots, a_{N}$ ãïŒä»»æã® $1$ ä»¥äž $N$ 以äžã®æŽæ° $n$ ã«ã€ããŠä»¥äžã®åŒãæºãããŠããŸãïŒ
$$\bigg\lfloor \frac{2n}{3} \bigg\rfloor = \sum_{k = 1}^{n} a_k \bigg\lfloor \frac{n}{k} \bigg\rfloor$$
ã$S = \\{a_k\mid 1 \le k \le N\\}$ ãšãããšãïŒ$\sum_{a\in S}|a|$ ã解çããŠãã ããïŒããªãã¡ïŒ$a_1,\ldots, a_{N}$ ã®äžã«çŸããå®æ°ãã¹ãŠã«ã€ããŠïŒãã®çµ¶å¯Ÿå€ã®ç·åã解çããŠãã ããïŒ\
ããã ãïŒãã®ãã㪠$a_1, a_2, \ldots, a_{N}$ ã¯äžæã«å®ãŸãããšãä¿èšŒãããŸãïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/8619 | A | OMCB015(A) | 100 | 315 | 325 | [
{
"content": "ãåè§åœ¢ $ABCD$ ã®é¢ç©ã $S$ ãšãïŒ$AC$ ãš $BD$ ã®äº€ç¹ã $X$ ãšãïŒ$\\angle{AXB}=\\theta$ ãšãããšïŒ\r\n$$\r\n\\begin{aligned}\r\nS&=\\dfrac{1}{2}\\bigl(AX\\cdot BX\\sin\\theta+BX\\cdot CX\\sin\\(\\pi-\\theta)+CX\\cdot DX\\sin\\theta+DX\\cdot AX\\sin(\\pi-\\theta)\\bigr)\\\\\\\\\r\n&=\\dfrac{1}{2}(AX\\cdot BX+BX\\cdot CX+CX\\cdot DX+DX\\cdot AX)\\sin\\theta \\\\\\\\\r\n&=\\dfrac{1}{2}(AX+CX)(BX+DX) \\sin\\theta \\\\\\\\\r\n&=\\dfrac{1}{2}AC\\cdot BD \\sin\\theta\r\n\\end{aligned}\r\n$$\r\nã§ããããïŒ$\\sin\\theta\\leq1$ ããã³çžå ã»çžä¹å¹³åã®äžçåŒã«ãã\r\n$$\r\nS\\leq \\dfrac{1}{2}AC\\cdot BD\\leq \\dfrac{1}{2}\\biggl(\\frac{AC+BD}{2}\\biggr)^2=1250.\r\n$$\r\nçå·ã¯ $AC\\perp BD$ ã〠$AC=BD=50$ ã®ãšãæç«ããããïŒæ±ããçã㯠$\\mathbf{1250}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/8619"
}
] | ãåžåè§åœ¢ $ABCD$ ã«ãããŠïŒ$AC+BD=100$ ãæãç«ã€ãšãïŒåè§åœ¢ $ABCD$ ã®é¢ç©ãšããŠããããæ倧å€ãæ±ããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/11333 | B | OMCB015(B) | 200 | 262 | 282 | [
{
"content": "ããŸãïŒ $z=y^2+2y+3$ ãšãããšïŒ\r\n $$\r\nz=(y+1)^2+2\r\n $$\r\nããïŒ $z\\geq 2$ ãæºããïŒ$f(x,y)=0$ ã®è§£ã¯ $x=-z\\pm \\sqrt{z^2+4}$ ãšãªãã®ã§ïŒå€§ããæ¹ã®è§£ã«ã€ããŠèãããšïŒ\r\n $$\r\nx=-z+\\sqrt{z^2+4}=\\dfrac{4}{z+\\sqrt{z^2+4}}\r\n $$\r\nãšãªãã®ã§ïŒ $z+\\sqrt{z^2+4}$ ã®æå°å€ãèããã°ããïŒ\r\nããã§ïŒ $z+\\sqrt{z^2+4}$ 㯠$z\\gt0$ ã§å調å¢å ãªé¢æ°ã§ããã®ã§ïŒ$z=2$ ã§æå°å€ $2+2\\sqrt2$ ãåãïŒ\\\r\nããã£ãŠïŒ $y=-1$ ã®ãšã $x$ ã¯æå€§å€ $-2+\\sqrt8$ ããšãã®ã§ïŒè§£çãã¹ãå€ã¯ $\\mathbf{16}$ ãšãªãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/11333"
},
{
"content": "ã$f(x, y) = 2xy^2 + 4xy + x^2 + 6x - 4$ ã§ããïŒ$f(x, y) = 0$ ãæºããå®æ° $y$ ãååšããããã® $x$ ã«é¢ããæ¡ä»¶ã¯ïŒå€å¥åŒãèããããšã«ãã\r\n$$-8x^3 + 16x^2 - 48x^2 + 32x \\geq 0 \\Longleftrightarrow x(x^2 + 4x - 4) \\leq 0$$\r\nãšåããïŒ$x \\gt 0$ ã®å ŽåïŒãã㯠$x^2 + 4x - 4 \\leq 0$ ãšåå€ã§ããïŒãããæºããæ倧ã®å®æ° $x$ 㯠$2\\sqrt{2} - 2$ ã§ããïŒ$x \\leq 0$ ã®å ŽåïŒæããã«ããããå°ããå€ãã $x$ ã¯åãåŸãªãããïŒæ±ããã¹ãæ倧å€ã¯ $2\\sqrt{2} - 2$ ãšåããïŒ",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/11333/560"
}
] | ãå®æ° $x , y$ ã«å¯ŸããŠé¢æ° $f(x , y)$ ã以äžã®ããã«å®çŸ©ããŸãïŒ
$$
f(x , y)=x^2+2(y^2+2y+3)x-4
$$
$f(x , y)=0$ ãæºãããªãã $x,y$ ãåããšãïŒ$x$ ã®æ倧å€ãæ±ããŠãã ããïŒãã ãæ±ããå€ã¯æ£ã®æŽæ° $a , b$ ãçšã㊠$\sqrt{a}-b$ ãšè¡šããã®ã§ïŒ $ab$ ã®å€ã解çããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/3622 | C | OMCB015(C) | 200 | 186 | 265 | [
{
"content": "ã$10^{3622}-1$ 以äžã®æ£æŽæ°ã§ãã£ãŠïŒ$10^{3621}$ ã®äœã $1$ ã§ãããã®ïŒ$10^{3620}$ ã®äœã $1$ ã§ãããã®ïŒ$\\cdots$ ïŒ$10^{0}$ ã®äœã $1$ ã§ãããã®ã¯ïŒãããããã¹ãŠ $10^{3621}$ åã§ããïŒããã $2,3,\\cdots ,9$ ã§ãåæ§ã«èããããšã§ïŒ\r\n$$\r\nS=(10^{3621}\\times 3622)(1+2+\\ldots +9)+1=16299 \\underbrace{0\\cdots 0}_{3621å}1\r\n$$ \r\nãããã£ãŠïŒè§£çãã¹ãå€ã¯ $\\textbf{28}$ ã§ããïŒ\r\n\r\n----\r\n**å¥è§£ïŒ**\\\r\nã$10^{3622}-1$ 以äžã®éè² æŽæ° $10^{3622}$ åã«ã€ããŠïŒåæ¡ã®æåŸ
å€ã¯ $4.5$ ãªã®ã§ïŒæ¬¡ã®ããã« $S$ ãæ±ããããšãã§ããïŒ\r\n$$S=4.5\\times 3622\\times 10^{3622}+1=16299 \\underbrace{0 \\cdots 0} _{3621å}1$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/3622"
}
] | ã$10^{3622}$ 以äžã®æ£æŽæ°ãã¹ãŠã«ã€ããŠïŒããããã®åæ¡ã®åã®ç·åã $S$ ãšããŸãïŒ$S$ ã®åæ¡ã®åãæ±ããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/4546 | D | OMCB015(D) | 200 | 193 | 264 | [
{
"content": "ã$f(n)=180^\\circ-\\dfrac{360^\\circ}{n}$ ããïŒæ¡ä»¶ã¯æ¬¡ãšåå€ã§ããïŒ\r\n$$\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}=\\frac{1}{2},\\quad 3\\leq a\\lt b\\lt c$$\r\nç¹ã« $\\dfrac{3}{a}\\gt \\dfrac{1}{2}$ ããïŒ$a=3,4,5$ ã§ããïŒ\r\n- $a=3$ ã®ãšãïŒæ¡ä»¶ã¯æ¬¡ãšåå€ãªã®ã§ $c=15,18,24,42$ ãåŸãïŒ\r\n$$(b-6)(c-6)=36,\\quad 4\\leq b\\lt c$$\r\n- $a=4$ ã®ãšãïŒæ¡ä»¶ã¯æ¬¡ãšåå€ãªã®ã§ $c=12,20$ ãåŸãïŒ\r\n$$(b-4)(c-4)=16,\\quad 5\\leq b\\lt c$$\r\n- $a=5$ ã®ãšãïŒæ¡ä»¶ã¯æ¬¡ãšåå€ã ãïŒãããæºãã $(b,c)$ ã®çµã¯ååšããªãïŒ\r\n$$(3b-10)(3c-10)=100,\\quad 6\\leq b\\lt c$$\r\n\r\nã以äžããïŒ$c$ ãšããŠããåŸãå€ã®ç·å㯠$\\bf131$ ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/4546"
}
] | ã$3$ 以äžã®æŽæ° $n$ ã«å¯ŸãïŒåºŠæ°æ³ã§ã®æ£ $n$ è§åœ¢ã®äžã€ã®å
è§ã®å€§ããã $f(n)$ ã§è¡šããŸãïŒ$3\leq a\lt b\lt c$ ãªãæŽæ°ã®çµ $(a,b,c)$ ã $f(a)+f(b)+f(c)=360^\circ$ ãã¿ãããšãïŒ$c$ ãšããŠããããå€ã®ç·åãæ±ããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/5163 | E | OMCB015(E) | 200 | 199 | 225 | [
{
"content": "ã蟺 $AD$ äžã« $AE = 5$ ãªãç¹ $E$ ãåããšïŒ$DE = 3$ ã§ããããäžè§åœ¢ $CDE$ ã¯æ£äžè§åœ¢ã§ããïŒåŸã£ãŠ $CE = 3$ ã§ããããïŒäžè§åœ¢ $ABC$ ãš $AEC$ ã¯äžèŸºçžçã§ååã§ããïŒåŸã£ãŠ $\\angle ABC = \\angle AEC = 120^\\circ$ ãåããã®ã§ïŒäœåŒŠå®çãã $AC = 7$ ãåããïŒãŸãïŒ\r\n$\\angle ABC + \\angle ADC = 180^\\circ$\r\nã§ããããåè§åœ¢ $ABCD$ ã¯åã«å
æ¥ããã®ã§ïŒPtolemy ã®å®çããæ±ããå€ã¯\r\n$$BD = \\frac{AB\\times CD + BC\\times DA}{AC} = \\frac{39}{7}$$\r\nãšèšç®ã§ããïŒç¹ã«è§£çãã¹ãå€ã¯ $\\bf{46}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/5163"
}
] | ãåžåè§åœ¢ $ABCD$ ã¯ïŒ
$$AB=5,\quad BC=CD=3,\quad DA=8,\quad \angle CDA = 60^\circ$$
ãæºãããŸãïŒ$BD$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šããããïŒ$a+b$ ã解çããŠäžããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/8425 | F | OMCB015(F) | 200 | 172 | 210 | [
{
"content": "ãå€å¥åŒãèããããšã§ïŒæ¡ä»¶ã¯ä»¥äžã®ããã«è¡šããïŒ\r\n$$ a+b=1, \\quad a^2\\geq 4b, \\quad b^2\\geq 4a.$$\r\n$ab$ å¹³é¢äžã§ããããå³ç€ºããããšã§ïŒ$a+b=1$ ã®ããšã§ããã¯ä»¥äžãšåå€ã§ãããšãããïŒ\r\n$$ a\\leq -2-2\\sqrt{2} \\quad \\text{ãŸãã¯} \\quad a\\geq 3+2\\sqrt{2}. $$\r\nããã« $ab$ å¹³é¢äžã§ååš $a^2+b^2=k$ ãšå
±éç¹ããã€æ¡ä»¶ãèããã°ïŒ$k$ 㯠$\\\\{a,b\\\\}=\\\\{-2-2\\sqrt{2},3+2\\sqrt{2}\\\\}$ ã®ãšãæå°å€ $ 29+20\\sqrt{2}$ ããšãïŒç¹ã«ïŒè§£çãã¹ãå€ã¯ $29+20+2=\\mathbf{51}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/8425"
}
] | ãå®æ° $a,b$ 㯠$a+b=1$ ãã¿ãããŠããŸãïŒãŸãïŒ$x$ ã«ã€ããŠã® $2$ 次æ¹çšåŒ
$$x^2+ax+b=0,\quad x^2+bx+a=0$$
ã¯ïŒããããå°ãªããšãäžã€ã®å®æ°è§£ãæã¡ãŸãïŒãã®ãšãïŒ$a^2+b^2$ ã®ãšãããæå°å€ã¯ïŒæ£ã®æŽæ° $A,B,C$
ïŒ $C$ ã¯å¹³æ¹å åãæããªãïŒãçšã㊠$A+B\sqrt{C}$ ãšè¡šãããã®ã§ïŒ$A+B+C$ ã解çããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/10781 | G | OMCB015(G) | 300 | 121 | 191 | [
{
"content": "ãé¢ã®åæ°ã $f$ ãšããïŒæãå€åŽã®éããé¢ãå«ãããšã«æ³šæããïŒãšïŒãªã€ã©ãŒã®å€é¢äœå®çããïŒ\r\n$$\r\nn=10000-f+2\r\n$$\r\nãæãç«ã€ïŒãã£ãŠ $f$ ãæ倧ã§ãããšããèããã°ããïŒåé¢ã¯ $3$ 蟺以äžæã€ããïŒæ¬¡ãæãç«ã€ïŒ\r\n$$\r\n3f \\leqq 2 \\cdot 10000\r\n$$\r\nãã£ãŠ $f$ ã®æ倧å€ã¯ $6666$ ã§ããïŒãã®ãšã $n = 3336$ ãšãªãïŒå®éïŒæ¬¡ã®ããã« $3336$ åã®ç¹ããšã£ãŠïŒå $X_k$ ãš $A,B$ ãçµã³ïŒç·å $X_iX_{i+1}(i=1,2,...,3332)$ ãçµã¶ãšã㧠$10000$ æ¬ã®ç·åãåŒãããšãã§ããïŒ\r\n$$A(-1,0),~ B(1,0),~ X_{k}(0,k) \\quad (k=1,2,...,3334)$$\r\n以äžãã $n$ ã®æå°å€ã¯ $\\mathbf{3336}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/10781"
},
{
"content": "åžå
ã $k$ ç¹ãããªããšããŸãïŒ $n$ ç¹ã®ãã¡ãã $3$ ç¹ãååšããŠããããå
±ç·ãšãªããããªå Žåã¯æããã«ç¡é§ãªã®ã§ç¡èŠããŸãïŒ(ããã«ããïŒ $n$ ãåå倧ããæ $k \\geq 3$ ã«æ³šæ)\r\n\r\näžè§åœ¢åå²ããåŸã®å
è§ã®ç·åãèããŸãïŒ\r\n\r\nåžå
ãæ§æããé ç¹ã¯å
šéšåãã㊠$k$ è§åœ¢ã®å
è§ã®ç·åïŒããªãã¡ $180k-360$ 床éãŸãïŒãã以å€ã®é ç¹ã«ã¯ãããã $360$ 床éãŸãã®ã§ïŒåèšãããš $360(n-k)+180k-360=360n-360-180k$ 床ãšãªããŸãïŒ\r\n\r\nããªãã¡ïŒäžè§åœ¢ã¯ $2n-2-k$ åéãŸããŸãïŒ\r\n\r\näžè§åœ¢ã®èŸºã¯ïŒåžå
ã®èŸºä»¥å€ã¯ããã«ã«ãŠã³ãããïŒåžå
ã®èŸºã¯äžåã ãã«ãŠã³ããããããšããïŒèŸºã®æ°ã¯ $3n-3-k$ æ¬åŒããïŒããã $10000$ 以äžãšãªãæå°ã® $n$ 㯠$3336$ ãšåãããŸãïŒ",
"text": "åžå
ã䜿ããã€",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/10781/565"
}
] | ã$n$ ã $2$ 以äžã®æ£æŽæ°ãšããŸãïŒå¹³é¢äžã«çžç°ãªã $n$ åã®ç¹ããšãïŒ**è¯ãç¹**ãšããŸãïŒçžç°ãªã $2$ ã€ã®è¯ãç¹ãçµã¶ç·åãïŒæ¬¡ãæãç«ã€ããã« $10000$ æ¬åŒãããšãã§ããŸããïŒ
- $10000$ æ¬ã®ç·åã®ãã¡ïŒã©ã® $2$ æ¬ã端ç¹ãé€ããŠå
±æç¹ãæããªãïŒ
$n$ ãšããŠããããæå°å€ãæ±ããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/9335 | H | OMCB015(H) | 300 | 68 | 105 | [
{
"content": "ã$v_{2}(n)v_{3}(n)$ ã¯æ¬¡ã®ããã«èšããããããïŒ\r\n\r\n- $n$ ãå²ãåã $2$ 以äžã® $2$ ã¹ã $2^a$ ãš $3$ 以äžã® $3$ ã¹ã $3^b$ ã®ã㢠$(2^a,3^b)$ ã®åæ°ïŒ\r\n\r\n$(2^a,3^b)$ ãåºå®ãããšãïŒåæ¹ã§å²ããïŒããªãã¡ $2^a3^b$ ã§å²ããïŒ$3^6+1\\leq N \\leq 3^7$ ã®åæ°ã¯ïŒ\r\n$$ \\left \\lfloor \\dfrac{3^7}{2^{a}3^{b}} \\right \\rfloor - \\left \\lfloor \\dfrac{3^6}{2^{a}3^{b}} \\right \\rfloor.$$\r\nã§äžããããããïŒãã¹ãŠã® $(a, b)$ ã«å¯ŸããŠããã®ç·åããšãã°ããïŒ\r\n$$ \\left ( \\sum_{a = 1}^{\\infty} \\left \\lfloor \\dfrac{3^6}{2^{a}} \\right \\rfloor + \\sum_{a = 1}^{\\infty} \\left \\lfloor \\dfrac{3^5}{2^{a}} \\right \\rfloor + \\cdots + \\sum_{a = 1}^{\\infty} \\left \\lfloor \\dfrac{1}{2^{a}} \\right \\rfloor \\right ) - \\left ( \\sum_{a = 1}^{\\infty} \\left \\lfloor \\dfrac{3^5}{2^{a}} \\right \\rfloor + \\sum_{a = 1}^{\\infty} \\left \\lfloor \\dfrac{3^4}{2^{a}} \\right \\rfloor + \\cdots + \\sum_{a = 1}^{\\infty} \\left \\lfloor \\dfrac{1}{2^{a}} \\right \\rfloor \\right )$$\r\nãšå€åœ¢ããããšã§ïŒ$\\displaystyle \\sum_{a = 1}^{\\infty} \\left \\lfloor \\dfrac{3^6}{2^{a}} \\right \\rfloor = \\mathbf{723}$ ãšæ±ããããïŒãªãïŒæåŸã®ç·å㯠$v_2(3^6!)$ ã«çããïŒãã㯠$3^6$ ãã $3^6$ ã®äºé²æ³è¡šèšã§ã®æ¡åãæžããããšã§æ±ããããããšãç¥ãããŠããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/9335"
},
{
"content": "ãäžè¬ã«ïŒçžç°ãªãçŽ æ° $p,q$ ããã³éè² æŽæ° $n$ ã«å¯ŸããŠ\r\n$$\\sum_{N=p^n+1}^{p^{n+1}}v_p(N)v_q(N)=v_q(p^n!)$$\r\nãæãç«ã€ããšã $n$ ã«é¢ããæ°åŠçåž°çŽæ³ã§ç€ºããïŒ\r\n\r\n- $n=0$ ã®ãšãã¯äž¡èŸºãšãå€ã¯ $0$ ãšãªãçåŒãæç«ããïŒ\r\n- ããéè² æŽæ° $n$ ã§çåŒãæç«ãããšä»®å®ããïŒ$p\\nmid N$ ã®ãšã $v_p(N)=0$ ã§ããããšã«æ³šæãããšïŒ\r\n$$\\begin{aligned}\r\n\\sum_{N=p^{n+1}+1}^{p^{n+2}}v_p(N)v_q(N)&=\\sum_{N=p^{n}+1}^{p^{n+1}}v_p(pN)v_q(pN)\\\\\\\\\r\n&=\\sum_{N=p^{n}+1}^{p^{n+1}}(v_p(N)+1)v_q(N)\\\\\\\\\r\n&=\\sum_{N=p^{n}+1}^{p^{n+1}}v_p(N)v_q(N)+\\sum_{N=p^{n}+1}^{p^{n+1}}v_q(N)\\\\\\\\\r\n&=v_q(p^n!)+v_q\\Big(\\frac{p^{n+1}!}{p^n!}\\Big)\\\\\\\\\r\n&=v_q(p^{n+1}!)\r\n\\end{aligned}$$\r\nããïŒ$n+1$ ã®ãšããçåŒãæç«ããïŒ\r\n\r\nãã£ãŠç€ºãããïŒ\\\r\nãç¹ã«æ¬å㯠$p=3,q=2,n=6$ ã®å Žåã§ããïŒ",
"text": "ãªãv_2(3^6!)ã«çãããïŒ",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/9335/558"
},
{
"content": "$$\\begin{aligned}\r\n\\sum_{n=1}^{3^7} v_2(n) v_3(n) &= \\sum_{n=1}^{3^6} (v_2(3n-2) v_3(3n-2) + v_2(3n-1)v_3(3n-1) + v_2(3n) v_3(3n))\\\\\\\\\r\n&= \\sum_{n=1}^{3^6} (0 + 0 + v_2(n) (v_3(n) + 1))\\\\\\\\\r\n&= \\sum_{n=1}^{3^6} v_2(n) v_3(n) + \\sum_{n =1}^{3^6} v_2(n)\r\n\\end{aligned}$$\r\nãã\r\n$$\\begin{aligned}\r\n\\sum_{n= 3^6 + 1}^{3^7} v_2(n) v_3(n) &= \\sum_{n=1}^{3^6} v_2(n)\\\\\\\\\r\n&= \\sum_{k=1}^{\\infty} \\left( \\left\\lfloor \\frac{3^6}{2^k} \\right\\rfloor - \\left\\lfloor \\frac{3^6}{2^{k+1}} \\right\\rfloor \\right) \\times k\\\\\\\\\r\n&= \\sum_{k=1}^{\\infty} \\left\\lfloor \\frac{3^6}{2^k} \\right\\rfloor\\\\\\\\\r\n&= \\bf{723}\r\n\\end{aligned}$$\r\nãšãªãïŒ",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb015/editorial/9335/563"
}
] | ãæ£æŽæ° $n$ ãçŽ æ° $p$ ã§å²ãåããæ倧ã®åæ°ã $v_{p}(n)$ ã§è¡šããšãïŒ
$$\displaystyle \sum_{n = 3^6+1}^{3^7} v_{2}(n)v_{3}(n)$$
ãæ±ããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/5029 | A | OMCB014(A) | 100 | 300 | 304 | [
{
"content": "ã$n=1$ ã¯é©ããïŒ$n=2,3,4$ ã¯é©ããïŒ$n\\ge5$ ã®ãšãïŒ$5$ çªç®ã®çŽ æ°ã $9$ ãã倧ããããšãšïŒ$n+1$ çªç®ã®çŽ æ°ã $n$ çªç®ã®çŽ æ°ãã $2$ 以äžå€§ããããšããïŒã€ãã«é©ããªãïŒãã£ãŠïŒæ±ããç·å㯠$2+3+4=\\mathbf{9}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/5029"
}
] | ã$n$ çªç®ã«å°ããçŽ æ°ã $2n-1$ ã§ãããããªïŒæ£æŽæ° $n$ ã®ç·åã解çããŠäžããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/6183 | B | OMCB014(B) | 100 | 278 | 293 | [
{
"content": "ã$a=2x+1, b=2y+1, c=2z+1$ ãšããããšã§ïŒæ¡ä»¶ãæºããçµã®åæ°ã¯ $x+y+z=49$ ãæºããé åºä»ããéè² æŽæ°ã®çµ $(x,y,z)$ ã®åæ°ã«çããããšããããïŒãã®åæ°ã¯ïŒ$49$ åã®ããŒã«ãš $2$ åã®ä»åãã䞊ã¹ãæ¹æ³ã®åæ°ã«äžèŽããã®ã§ïŒæ±ããçã㯠${}\\_{49+2}\\mathrm{C}\\_{2}=\\textbf{1275}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/6183"
}
] | ã$a+b+c=101$ ãæºããæ£ã®å¥æ°ã®çµ $(a, b, c)$ ã¯ããã€ååšããŸããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/9747 | C | OMCB014(C) | 100 | 282 | 291 | [
{
"content": "ãOMCãããæåã«æãæµ®ãã¹ãŠããæ°ã $n$ ãšãããšïŒOMCãããæãæµ®ãã¹ãŠããæ°ã¯æ¬¡ã®ããã«å€åããïŒ\r\n\r\n$$\\begin{aligned}\r\nn&\\longrightarrow 10n \\longrightarrow 10n+a \\longrightarrow 110n+11a \\longrightarrow 110n + 11a - b \\\\\\\\\r\n&\\longrightarrow \\dfrac{110n+11a-b}{a} \\longrightarrow \\dfrac{(110-a)n+11a-b}{a}\r\n\\end{aligned}$$\r\næçµçãªèšç®çµæã $n$ ã«ãããªããšãïŒ$110-a=0$ ã§ããã®ã§ $a=110$ ïŒèšç®çµæ㯠$\\dfrac{1210-b}{110}$ ãšãªãïŒããã $1$ ãšçããã®ã§ $b=1100$ ãåŸãïŒãããã£ãŠ $a+b=\\bf1210$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/9747"
}
] | ãOMC ãããããæ£æŽæ°ãäžã€æãæµ®ãã¹ãŠããŸãïŒããã§ïŒããªãã¯æ£æŽæ° $a,b$ ãçšããŠä»¥äžã®ãããªèšç®ãããããã« OMC ããã«æ瀺ãããŸãïŒ
- æãæµ®ãã¹ãŠããæ°ã $10$ åã㊠$a$ ã足ãïŒããã $11$ åããŠãã $b$ ãåŒããŠïŒããã«ããã $a$ ã§å²ã£ãŠããæåã«æãæµ®ãã¹ãŠããæ°ãåŒãïŒ
ãã®ãšãïŒæçµçãªèšç®çµæã¯ïŒOMC ãããæåã«æãæµ®ãã¹ãŠããæ°ã«ããã $1$ ãšãªããŸãïŒ$a+b$ ã®å€ãæ±ããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/5732 | D | OMCB014(D) | 200 | 215 | 259 | [
{
"content": "ãæ±ããååŸã $r$ ãšããïŒãã®åã®äžå¿ã $O$ ãšããïŒãã®ãšã, $O$ ãã蟺 $CD$ ã«åç·ããããïŒãã®è¶³ã $H$ ãšãããšïŒ\r\n$$OD = r + \\frac{1}{3}, \\quad OH = \\frac{1}{2}, \\quad DH = 1 - r$$\r\nã§ããããïŒäžè§åœ¢ $DOH$ ã«ã€ããŠäžå¹³æ¹ã®å®çãã\r\n$$\\left(r + \\frac{1}{3} \\right)^2 = (1 - r)^2 + \\frac{1}{4}$$\r\nããã解ãããšã§ $r = \\dfrac{41}{96}$ ãåŸããïŒè§£çãã¹ãå€ã¯ $\\mathbf{137}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/5732"
}
] | ãäžèŸºã®é·ãã $1$ ã®æ£æ¹åœ¢ $ABCD$ ã«ã€ããŠïŒç¹ $A, D$ ãããããäžå¿ãšãïŒååŸããšãã« $\dfrac{1}{3}$ ã® $2$ ã€ã®åãèããŸãïŒãã® $2$ ã€ã®åã«å€æ¥ãïŒèŸº $BC$ ãšãæ¥ããåã®ååŸãæ±ããŠãã ããïŒãã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a, b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/4550 | E | OMCB014(E) | 200 | 251 | 269 | [
{
"content": "$$(\\text{äžåŒ}) \\iff2024\\leq\\dfrac{10^n}{m}\\lt2025\\iff\\dfrac{1}{2025}\\lt\\dfrac{m}{10^n}\\leq\\dfrac{1}{2024}$$\r\nãã㧠$\\dfrac{1}{2025}=0.000493\\cdots$ ããã³ $\\dfrac{1}{2024}= 0.000494\\cdots$ ããïŒæ±ããæå°ã® $m$ 㯠$\\bf{494}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/4550"
}
] | ã$\bigg\lfloor \dfrac{10^n}{m} \bigg\rfloor=2024$ ãæºããæ£æŽæ°ã®çµ $(m,n)$ ã®ãã¡ïŒ$m$ ãæå°ã§ãããã®ã«ã€ããŠïŒ$m$ ã®å€ãæ±ããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/6090 | F | OMCB014(F) | 300 | 227 | 235 | [
{
"content": "ã$a_n+a_{n+1} \\gt a_na_{n+1}$ ãšãªãã®ã¯ïŒ$a_n$ ãš $a_{n+1}$ ã®å°ãªããšãçæ¹ã $1$ ãšãªããšãã«éãããïŒåŸã£ãŠïŒ$\\lbrace a_n \\rbrace $ ã¯ä»¥äžã®ããã«ãªããšããã. \\\r\n$$a_1=a_2=1,\\quad a_3=2,\\quad a_4=3,\\quad a_{n+2}=a_na_{n+1}\\quad (n=3,4,\\ldots)$$\r\nãããã§ïŒãã£ããããæ°åã®ç¬¬ $n$ é
ã $F_{n}$ ãšè¡šãããšã«ããïŒã€ãŸãïŒæ¬¡ã®ããã«æ°å $\\\\{F_n\\\\}$ ãå®ããïŒ\r\n$$F_{-1}=1,\\quad F_{0}=0,\\quad F_{n+2}=F_{n+1}+F_{n}$$\r\nãã®ãšãïŒåž°çŽçã« $n\\geq 3$ ã§ã¯ $a_n = 2^{F_{n-4}}\\times 3^{F_{n-3}}$ ã§ããããšã確ãããããïŒãã£ãŠïŒ$a_n$ ã®çŽæ°ã®åæ°ã¯ $(F_{n-4}+1)(F_{n-3}+1)$ ã§ããããïŒ$k=14$ ãåããïŒ$a_{14}=2^{55}\\times3^{89}$ ãªã®ã§ïŒç¹ã«è§£çãã¹ãå€ã¯ $\\bf{29370}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/6090"
}
] | ãæ°å$\lbrace a_n \rbrace$ ã以äžã®ããã«å®ããŸã. \
$$
a_1=a_2=1,\quad a_{n+2}=\max (a_n+a_{n+1} , a_na_{n+1})\quad(n=1,2,\ldots)
$$
ãã®ãšãïŒæ£ã®çŽæ°ã $5040$ åæã€é
$a_k$ ãã¡ããã©äžã€ååšããŸãïŒ$a_k$ ã¯çŽ æ° $p,q$ ãšæ£ã®æŽæ° $a,b$ ãçšã㊠$a_k=p^a\times q^b \space$ ãšè¡šããã®ã§ïŒ$abpq$ ã解çããŠãã ãã. |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/4319 | G | OMCB014(G) | 300 | 183 | 242 | [
{
"content": "ã$N=\\overline{abc}$ ã«ã€ããŠïŒ$a\\lt b\\lt c$ ã§ãããšããŠãäžè¬æ§ã倱ããªãïŒããŸïŒ$b-a$ ãš $c-b$ ã®å°ãªããšãäžæ¹ã¯ $4$ 以äžã§ããããïŒ$\\overline{cba}-\\overline{cab}=9(b-a)$ ãš $\\overline{acb}-\\overline{abc}=9(c-b)$ ã®å°ãªããšãäžæ¹ã¯ $36$ 以äžã§ããïŒããªãã¡ïŒ$m$ ã¯ã$36$ 以äžã® $9$ ã®åæ°ããåæ°ã«ãã€ïŒ\\\r\nãããŠïŒ$m$ ã $4$ ã®åæ°ã§ãããšãããšïŒ$a,b,c\\in\\\\{2,4,6,8\\\\}$ ã§ãªããã°ãªããªããïŒãã®ãšãæ¡ä»¶ãã¿ããåŸãªãããšã確èªã§ããïŒãŸãïŒ$m=27$ ãäžé©ã§ããããšããããïŒ\\\r\nãéã« $N=468$ ãšããã°ïŒ$18$ ã®çŽæ°ã¯ãã¹ãŠé©ããããïŒãããã®ç·å㯠$\\mathbf{39}$ ã§ããïŒ\\\r\nããªãïŒåäœã®æ°ãšã㊠$0$ ã蚱容ããã°ïŒ$N=(0)48$ ãšããã° $m=4$ ãš $m=12$ ãè¿œå ã§é©ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/4319"
}
] | ãåäœã®æ°ãçžç°ãªãïŒã〠$0$ ãå«ãŸãªã $3$ æ¡ã®æ£æŽæ° $N$ ããããŸãïŒ$N$ ã®åäœã®æ°ã䞊ã¹æ¿ããŠåŸããã $6$ åã®æ£æŽæ°ïŒ$N$ ãå«ãïŒã¯ãã¹ãŠ $m$ ã®åæ°ã§ããïŒãã®ãšãïŒ$m$ ãšããŠããåŸãæ£æŽæ°ã®ç·åãæ±ããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/7159 | H | OMCB014(H) | 300 | 104 | 146 | [
{
"content": "ãåè§åœ¢ $ABCD, BCDE, CDEA$ ã¯ããããçèå°åœ¢ã§ããïŒãã£ãŠïŒ\r\n$$\\angle ACD = \\angle CAE = \\angle CBE = \\angle BED = \\angle BAD = \\angle ADC$$\r\nã§ãããã $AC = AD$ ã§ããïŒãŸãïŒ$AC = BD, AD = CE$ ãæãç«ã€ã®ã§ïŒ\r\n$$AC = AD = BD = CE$$\r\nã§ããïŒãã®é·ãã $x$ ãšããïŒãã®ãšãïŒåè§åœ¢ $ABCD$ ã«å¯ŸããŠPtolemyã®å®çãé©çšããããšã§\r\n$4 + 3x = x^2$\r\nãåããã®ã§ïŒ$x = 4$ ãåŸãïŒ\\\r\nãããã§ïŒ$\\Gamma$ ã®äžå¿ã $O$ ãšãïŒèŸº $CD$ ã®äžç¹ã $M$ ãšããïŒãã®ãšãïŒäžè§åœ¢ $ACD$ ã¯éè§äºç蟺äžè§åœ¢ã§ãããã $O$ ã¯ç·å $AM$ äžããããšã«æ°ãã€ããïŒäžå¹³æ¹ã®å®çãã\r\n$AM = \\sqrt{x^2 - 1^2} = \\sqrt{15}$\r\nã§ããïŒãã£ãŠïŒ$OM = y$ ãšãããšïŒ\r\n$$\\sqrt{y^2 + 1} = OC = OA = \\sqrt{15} - y$$\r\nãæãç«ã€ã®ã§ïŒããã解ã㊠$y = \\dfrac{7}{\\sqrt{15}}$ ãåŸãïŒãããã£ãŠïŒ$\\Gamma$ ã®ååŸã¯\r\n$$OA = \\sqrt{15} - y = \\frac{8}{\\sqrt{15}}$$\r\nã§ããããïŒæ±ããé¢ç©ã¯ $\\dfrac{64}{15}\\pi$ ã§ããïŒç¹ã«ïŒè§£çãã¹ãå€ã¯ $\\bf79$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/7159"
},
{
"content": "ã匊 $AB$ ã«å¯Ÿããäžå¿è§ã $2x$ , 匊 $BC$ ã«å¯Ÿããäžå¿è§ã $2y$ ãšãããš, $3\\times(2x)+2\\times(2y)=2\\pi$ ãæç«. $\\Gamma$ ã®ååŸã $r$ ãšãããš, \r\n$$\\begin{aligned}\r\n\\sin x&=\\frac{\\frac{1}{2}AB}{r}=\\frac{1}{r}\\\\\\\\\r\n\\sin y&=\\frac{\\frac{1}{2}BC}{r}=\\frac{3}{2r}\r\n\\end{aligned}$$\r\nãæç«ã, $\\sin y$ ã« $y=\\dfrac{\\pi-3x}{2}$ ã代å
¥ãããš\r\n$$\\begin{aligned}\r\n\\sin x&=\\sin\\Big(2\\times\\frac{1}{2}x\\Big)\\\\\\\\\r\n&=2\\cos\\Big(\\frac{1}{2}x\\Big)\\sin\\Big(\\frac{1}{2}x\\Big)\\\\\\\\\r\n\\sin y&=\\cos\\Big(\\frac{3}{2}x\\Big)\\\\\\\\\r\n&=\\cos\\Big(\\frac{1}{2}x\\Big)\\Big(1-4\\sin^2\\Big(\\frac{1}{2}x\\Big)\\Big)\r\n\\end{aligned}$$\r\nãšãªã, $\\sin x :\\sin y=2:3$ ãã, $\\sin\\Big(\\dfrac{1}{2}x\\Big)=\\dfrac{1}{4}$ ãåŸãã, \r\n$$r=\\frac{1}{\\sin x}=\\frac{8}{\\sqrt{15}} $$\r\nãæç«ããã®ã§, $\\Gamma$ ã®é¢ç©ã¯ $\\pi r^2=\\dfrac{64}{15}\\pi$ ãšãªã.",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/7159/552"
}
] | ãå $\Gamma$ ã«å
æ¥ããåžäºè§åœ¢ $ABCDE$ ãïŒä»¥äžã®æ¡ä»¶ãæºãããŠããŸãïŒ
$$AB=CD=EA=2,\quad BC=DE=3$$
$\Gamma$ ã®é¢ç©ãæ±ããŠãã ããïŒãã ãïŒæ±ããé¢ç©ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}\pi$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/6554 | A | OMCE005(A) | 400 | 119 | 138 | [
{
"content": "ã$100$ ã®åçŽæ° $d$ ã® $f\\_{100}(100)$ ãžã®å¯äžãèããïŒ$f\\_{100}(100)$ ã«ãããŠïŒ$d$ 㯠\r\n$$a\\_{100}=100, \\quad a\\_{0}=d, \\quad a\\_{i}|a\\_{i+1}\\ (i=0,1,\\dots,99)$$ \r\nãæºããæŽæ°å $(a\\_{0},a\\_{1},\\dots,a\\_{100})$ ã®åæ°åã ãå ç®ãããïŒãã®æ°åã®åæ°ã¯åçŽ å æ°ããšã«ç¬ç«ã«èããããšãã§ãïŒ$d=2\\^{p}5^\\{q}$ ãšããã°ïŒäžèšã®æ°åã®åæ°ã¯ïŒ\r\n$$\\binom{101-p}{2-p}\\times\\binom{101-q}{2-q}$$\r\nåã§ããïŒãã£ãŠïŒæ±ããçã㯠\r\n$$\\begin{aligned}\r\nf\\_{100}(100)&=\\left(\\sum\\_{p=0}\\^{2}\\binom{101-p}{2-p}2\\^{p}\\right)\\left(\\sum\\_{q=0}\\^{2}\\binom{101-q}{2-q}5\\^{q}\\right) \\\\\\\\\r\n&=5254\\times5575 \\\\\\\\\r\n&=\\mathbf{29291050}\r\n\\end{aligned}$$\r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/6554"
}
] | ãæ£ã®æŽæ°ã«å¯ŸããŠå®çŸ©ããæ£ã®æŽæ°å€ããšãé¢æ°ã®å $\\{ f_i \\}$ ãïŒ$f_0(n) = n$ ããã³
$$ f\_i(n)= \sum\_{d\mid n}f\_{i-1}(d) \quad (i = 1, 2, 3, \ldots) $$
ã«ãã£ãŠå®ããŸãïŒãã®ãšãïŒ$f\_{100}(100)$ ãæ±ããŠãã ããïŒãã ãïŒ$\sum\limits\_{d\mid n}$ 㯠$n$ ã®ãã¹ãŠã®æ£ã®çŽæ° $d$ ã«ã€ããŠç·åããšãããšãæå³ããŸãïŒ |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/10902 | B | OMCE005(B) | 400 | 103 | 121 | [
{
"content": "ãæ±ããå€ã $S$ ãšããïŒ$f(n)$ ã¯ïŒæ£ã®æŽæ° $i$ ã§ãã£ãŠ $\\dfrac{n}{2^i}$ ã $2$ é²æ³è¡šèšãããšãã«å°æ°ç¬¬ $1$ äœã $1$ ãšãªããããªãã®ã®åæ°ïŒããªãã¡ $n$ ã $2$ é²æ³è¡šèšãããšãã«çŸãã $1$ ã®åæ°ã§ããïŒãããã£ãŠ $f(n)=f(2n)$ ãã $f(m)+f(2m)=f(m+2m)$ ãšãªããïŒ$2$ é²æ³è¡šèšã®å ç®ã§ç¹°ãäžãããçããå Žå $1$ ã®åæ°ã¯æžãã®ã§ïŒ$2$ é²æ³è¡šèšã§ $m+2m$ ã¯ç¹°ãäžãããçããªãïŒããªãã¡ $m$ ã® $2$ é²æ³è¡šèšã§ $1$ ãé£ç¶ããªãããšããããïŒéã« $m$ ã® $2$ é²æ³è¡šèšã§ $1$ ãé£ç¶ããªããšãïŒç¢ºãã« $f(3m)=2f(m)$ ãšãªãïŒä»¥äžããïŒ$1000$ æªæºã®æ£ã®æŽæ°ã§ $1$ ãé£ç¶ããªãæŽæ°ãè¯ãæŽæ°ãšåŒã¶ãšãïŒå
šãŠã®è¯ãæŽæ°ã«ã€ããŠã® $1$ ã®åæ°ã®ç·åã $S$ ã§ããïŒãã㧠$1001$ ä»¥äž $2^{10}$ æªæºã®æŽæ°ã¯ããããè¯ãæŽæ°ã§ã¯ãªãã®ã§ïŒ$2$ é²æ³ã§ $10$ æ¡ä»¥äžã®è¯ãæŽæ°ãèããã°ããïŒ\\\r\nã$2$ é²æ³ã§æé«äœã $0$ ã§ãªã $n$ æ¡ã®æ£æŽæ°ã®ãã¡ïŒè¯ãæŽæ°ã§ãããã®ã®åæ°ã $A_n$ ãšããïŒ$A_1=A_2=1$ ã§ããïŒè¯ãæŽæ°ã®æãäžã®æ¡ã $0$ ã $1$ ãã§å Žååããããšä»»æã®æ£ã®æŽæ° $n$ ã«ã€ã㊠$A_{n+2}=A_{n+1}+A_n$ ãåŸãïŒããªãã¡ $\\\\{ A_n \\\\}$ ã¯ãã£ããããæ°åãšäžèŽããïŒ\\\r\nããããã㯠$9$ æ¡ä»¥äžã®è¯ãæŽæ°ã¯ãã®é ã« $0$ ãè¿œå ããããšã§ $10$ æ¡ãšã¿ãªããŠèããïŒè¯ãæŽæ°ã®ãã¡äžãã $k$ æ¡ç®ã $1$ ã§ãããã®ã®åæ°ã $P_k$ $(1\\leq k\\leq 10)$ ãšããã°ïŒäžãã $1$ æ¡ç®ãã $k$ æ¡ç®ãŸã§ã® $k$ æ¡ã®éé ïŒããã³äžãã $k$ æ¡ç®ãã $10$ æ¡ç®ãŸã§ã® $11-k$ æ¡ããšãã«è¯ãæŽæ°ãšãªãããšããïŒ$P_k=A_kA_{11-k}$ ãšãªãïŒãããã£ãŠ \r\n$$S=\\displaystyle\\sum_{i=1}^{10}P_i =\\displaystyle\\sum_{i=1}^{10}A_iA_{11-i}$$ \r\nã§ããïŒãããèšç®ããã°\r\n$$S = 2\\cdot (1 \\cdot 55 + 1 \\cdot 34 + 2 \\cdot 21 + 3 \\cdot 13 + 5 \\cdot 8) = \\mathbf{420}$$\r\nãåŸãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/10902"
}
] | ãæ£ã®å®æ° $x$ ã«å¯Ÿã㊠$\lceil x \rfloor$ ãã $x$ ã®å°æ°ç¹ä»¥äžç¬¬äžäœãåæšäºå
¥ããŠåŸãããæŽæ°å€ããšå®çŸ©ããŸãïŒäŸãã°
$$\lceil 2 \rfloor=2, \quad \lceil 4.2 \rfloor=4, \quad \lceil 5.5 \rfloor=6$$
ãšãªããŸãïŒæ£ã®æŽæ° $n$ ã«å¯ŸããŠæŽæ° $f(n)$ ã
$$f(n)=\sum_{i=1}^{\infty} \left( \left\lceil \dfrac n{2^i} \right\rfloor-\left\lfloor \dfrac n{2^i} \right\rfloor \right)$$
ã§å®ãããšãïŒ$f(3m)=2f(m)$ ãæºãã $1000$ æªæºã®æ£ã®æŽæ° $m$ ãã¹ãŠã«ã€ã㊠$f(m)$ ã足ãåãããå€ã解çããŠãã ããïŒ |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/7261 | C | OMCE005(C) | 500 | 28 | 39 | [
{
"content": "ã$BC=x,CA=y,AB=z$ ãšãïŒçŽç· $DP$ ãš $BC$ ã®äº€ç¹ã $X$ïŒ$DQ$ ãš $CA$ ã®äº€ç¹ã $Y$ïŒ$DR$ ãš $AB$ ã®äº€ç¹ã $Z$ ãšãïŒåé¢äœ $DXYZ$ ã®äœç©ã $V_3$ ãšããïŒãŸãïŒç©ºéå
ã®ç¹ $W$ ãšçŽç· $t$ ã®è·é¢ã $d(W,t)$ ã§è¡šãïŒåã€ã®äžè§åœ¢ $ABC, DCB, CDA, BAD$ ã¯ååã§ããããšã«æ³šæããïŒ\\\r\nã$AX:BX = AD:BD = x : y$ ãªã©ãæç«ããããšããïŒ\r\n$$\\begin{aligned}\r\n\\frac{V_3}{V_1}\r\n&=\\frac{\\triangle{XYZ}}{\\triangle{ABC}}\\\\\\\\\r\n&=1-\\frac{x^2}{(x+y)(x+z)} - \\frac{y^2}{(y+z)(y+x)} - \\frac{z^2}{(z+x)(z+y)}\\\\\\\\\r\n&=\\frac{2xyz}{(x+y)(y+z)(z+x)}\r\n\\end{aligned}$$\r\nãšãªãïŒãŸãïŒ\r\n$$\\frac{DP}{DX} = 1 + \\frac{d(P,BC)}{d(D,BC)} = 1 + \\frac{2\\triangle{ABC}\\/(-x+y+z)}{2\\triangle{ABC}\\/x} = \\frac{y+z}{-x+y+z}$$\r\nãªã©ããïŒ\r\n$$\\dfrac{V_2}{V_3}=\\frac{DP}{DX}\\times\\frac{DQ}{DY}\\times\\frac{DR}{DZ}=\\frac{(y+z)(z+x)(x+y)}{(-x+y+z)(x-y+z)(x+y-z)}$$\r\nã§ããïŒåŸã£ãŠïŒ\r\n$$\\frac {V_2}{V_1}=\\frac{V_3}{V_1}\\times \\frac{V_2}{V_3} = \\frac {2xyz}{(-x+y+z)(x-y+z)(x+y-z)}$$\r\nã§ããïŒããã§ïŒäžè§åœ¢ $ABC$ ã®è§ $A,B,C$ å
ã®åæ¥åã®ååŸããããã $r_A,r_B,r_C$ ãšãããšïŒ\r\n$$r_A : r_B : r_C\r\n= \\frac{2\\triangle ABC}{-x+y+z} : \\frac{2\\triangle ABC}{x-y+z} : \\frac{2\\triangle ABC}{x+y-z}\r\n= \\frac{1}{-x+y+z} : \\frac{1}{x-y+z} : \\frac{1}{x+y-z}$$\r\nã§ããïŒãã£ãŠïŒ\r\n$$\\frac{V_2}{V_1}\r\n= \\frac{1}{4}r_Ar_Br_C\\bigg(\\frac{1}{r_B}+\\frac{1}{r_C}\\bigg)\\bigg(\\frac{1}{r_C}+\\frac{1}{r_A}\\bigg)\\bigg(\\frac{1}{r_A}+\\frac{1}{r_B}\\bigg)\r\n= \\frac{(r_B+r_C)(r_C+r_A)(r_A+r_B)}{4r_Ar_Br_C}$$\r\nã§ããïŒã㟠$\\\\{r_A, r_B, r_C \\\\} = \\\\{ 24, 25, 26 \\\\}$ ã§ããããïŒæ±ããæ¯ã¯ $\\dfrac {833}{416}$ ãšãªãïŒç¹ã«ïŒè§£çãã¹ãå€ã¯ $\\mathbf {1249}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/7261"
},
{
"content": "ãŸã,\r\n$$\r\nAD=BC=a, \\ BD=AC=b, \\ CD=AB=c\r\n$$\r\n$$\r\n\\overrightarrow{DA}=\\overrightarrow{a}, \\ \\overrightarrow{DB}=\\overrightarrow{b}, \\ \\overrightarrow{DC}=\\overrightarrow{c}\r\n$$\r\nãšãã.\r\n\r\nåå¿ã®äœçœ®ãã¯ãã«è¡šèšãã,\r\n$$\r\n\\overrightarrow{DP}=\\frac{c\\overrightarrow{b}+b\\overrightarrow{c}}{b+c-a}, \\ \r\n\\overrightarrow{DQ}=\\frac{c\\overrightarrow{a}+a\\overrightarrow{c}}{c+a-b}, \\ \r\n\\overrightarrow{DR}=\\frac{b\\overrightarrow{a}+a\\overrightarrow{b}}{a+b-c}\r\n$$\r\nãšè¡šãã.\r\n\r\nåé¢äœã®äœç©ã®å
¬åŒãã,\r\n$$\r\nV_{1}=\\frac{1}{6}\\left|\\left(\\overrightarrow{DA}\\times\\overrightarrow{DB}\\right)\\cdot\\overrightarrow{DC}\\right|, \\ \r\nV_{2}=\\frac{1}{6}\\left|\\left(\\overrightarrow{DP}\\times\\overrightarrow{DQ}\\right)\\cdot\\overrightarrow{DR}\\right|\r\n$$\r\nã§ãã, å€ç©ãšå
ç©ã®èšç®æ³åã«åŸã£ãŠæŽçããããšã§,\r\n$$\r\n\\frac{V_{2}}{V_{1}}=\\frac{2abc}{(b+c-a)(c+a-b)(a+b-c)}\r\n$$\r\nãåŸã.\r\n以éã¯[å
¬åŒè§£èª¬](https:\\/\\/onlinemathcontest.com\\/contests\\/omce005\\/editorial\\/7261)ã®éãåæ¥åã®ååŸã®æ§è³ªãçšãã.\r\n\r\n-----\r\n<details><summary>åå¿ã®äœçœ®ãã¯ãã«è¡šèš<\\/summary>\r\n<details><summary>åè: å
å¿ã®äœçœ®ãã¯ãã«è¡šèš<\\/summary>\r\n<\\/details>\r\n<\\/details>\r\n<details><summary>å€ç©ãšå
ç©ã®èšç®æ³å<\\/summary>\r\n\r\n空éãã¯ãã« $\\overrightarrow{x}, \\overrightarrow{y}, \\overrightarrow{z}$ ã«ã€ããŠ,\r\n- $\\overrightarrow{x} \\times \\overrightarrow{y} = -\\overrightarrow{y} \\times \\overrightarrow{x}$\r\n- $\\overrightarrow{x} \\parallel \\overrightarrow{y}$ ã®ãšã $\\overrightarrow{x} \\times \\overrightarrow{y} = \\overrightarrow{0}$\r\n- $\\left(\\overrightarrow{x} \\times \\overrightarrow{y}\\right) \\cdot \\overrightarrow{z} = \\left(\\overrightarrow{y} \\times \\overrightarrow{z}\\right) \\cdot \\overrightarrow{x} = \\left(\\overrightarrow{z} \\times \\overrightarrow{x}\\right) \\cdot \\overrightarrow{y}$\r\n\r\nãæãç«ã€.\r\n<\\/details>\r\n<details><summary>åæ¥åã®ååŸã®æ§è³ª<\\/summary>\r\näžè§åœ¢ $ABC$ ã®é¢ç©ã $S$, è§ $A, B, C$ ã«å¯Ÿããåæ¥åã®ååŸããããã $r_{A}, r_{B}, r_{C}$ ãšãããš,\r\n$$\r\n2S=r_{A}(CA+AB-BC)=r_{B}(AB+BC-CA)=r_{C}(BC+CA-AB)\r\n$$\r\n<details><summary>åè: å
æ¥åã®ååŸã®æ§è³ª<\\/summary>\r\näžè§åœ¢ $ABC$ ã®é¢ç©ã $S$, å
æ¥åã®ååŸã $r$ ãšãããš,\r\n$$\r\n2S=r(AB+BC+CA)\r\n$$\r\n<\\/details>\r\n<\\/details>",
"text": "空éãã¯ãã«",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/7261/556"
}
] | ãéè§äžè§åœ¢ $ABC$ ã® $3$ ã€ã®åæ¥åã®ååŸã¯ãããã $24,25,26$ ã§ãïŒç©ºéå
ã«ç¹ $D$ ã
$$AD=BC,\quad BD=AC,\quad CD=AB$$
ãæºããããã«åãïŒäžè§åœ¢ $DBC, DCA, DAB$ ã®è§ $D$ ã«å¯Ÿããåå¿ããããã $P,Q,R$ ãšããŸãïŒåé¢äœ $DABC,DPQR$ ã®äœç©ããããã $V_1,V_2$ ãšãããšãïŒ$\dfrac {V_2} {V_1}$ ã®å€ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac b a$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ãæ±ããŠäžããïŒ |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/3208 | D | OMCE005(D) | 700 | 29 | 58 | [
{
"content": "ã以éã¯ïŒäžã€ç®ã®æ¡ä»¶ãæºãããã®ïŒã€ãŸãïŒ`ã` ãš `ã` ã®éã« `()` ãšããé£ç¶æååããªããã®ã ããæååãšåŒã¶ããšã«ããïŒ\r\n\r\nãäžè¬ã«ïŒ`(` ãš `)` ãš `ã` ããããã $n$ åãã€ããå ŽåãèããïŒ$xy$ å¹³é¢äžã« $A(0,0)$, $B(0,1)$, $X(n,n)$, $Y(n,n+1)$ ããšãïŒ\r\n$x$ 軞æ£æ¹åãžã®é·ã $1$ ã®ç§»åã ã$\\to$ã ïŒ$y$ 軞æ£æ¹åãžã®é·ã $1$ ã®ç§»åã ã $\\uparrow$ ãã§è¡šãããšãšããïŒä»»æã®æååã«é¢ããŠïŒ\r\n- `ã` ãš `)` ã ããæãåºããŠå·Šããé ã« `ã` ã $\\to$ïŒ`)` ã $\\uparrow$ ã«å¯Ÿå¿ãããŠç¹ $A$ ããç¹ããŠãã£ããã®ã A-path\r\n- `ã` ãš `(` ã ããæãåºããŠå·Šããé ã« `ã` ã $\\to$ïŒ`(` ã $\\uparrow$ ã«å¯Ÿå¿ãããŠç¹ $B$ ããç¹ããŠãã£ããã®ã B-path\r\n\r\nãšåŒã¶ããšã«ããïŒäŸãã° $n=6$ ã®ãšã `ã(ã(ãã)((ã)))((ã))` ãè¡šã A-path,B-path ã¯äžå³ã®ããã«ãªãïŒ\r\n\r\n\r\n\r\nããã®ããã«ä»»æã®æååãç¢å°ã«å€æã㊠A-path ãäœããšïŒ`ã` ãš `)` 㯠$n$ åãã€ããã®ã§å§ç¹ãç¹ $A$ ãšãããšçµç¹ã¯å¿
ãç¹ $X$ ã«ãªãïŒåæ§ã« B-path ã¯å§ç¹ãç¹ $B$ ãšãããšïŒçµç¹ã¯ç¹ $Y$ ã«ãªãïŒéã«ä»»æã® A-path,B-path ã®çµããäžæã« $3n$ åã®æååã埩å
ã§ããïŒïŒããã¯éšåæåå `()` ãçŠæ¢ãããŠããããšããåŸãïŒããªãã¡ïŒåã $x$ 座æšã§ A-path ãš B-path ããããã« $\\uparrow$ ããããšã㯠`)` ãå
ã« A-path ã® $\\uparrow$ ã®æ°ã ã䞊ã¹ïŒãã®åŸã« `(`ã䞊ã¹ãïŒïŒ\r\n\r\n---\r\n\r\n**è£é¡ïŒ** æååãåã® $2$ ã€ã®æ¡ä»¶ãæºããããã®å¿
èŠååæ¡ä»¶ã¯ïŒA-path ãš B-path ãå
±æç¹ããããªãããšã§ããïŒ\r\n\r\n**蚌æïŒ** \r\nãå¿
èŠæ§ãæååã¯äžã€ç®ã®æ¡ä»¶ãæºãããŠããïŒæ¬åŒ§ã ãèŠããšæ£ããæ¬åŒ§åãªã®ã§ä»»æã® $x=iã(i=1,\\cdots,n)$ ã«ã€ããŠãã®æç¹ãŸã§ã«åºãŠãã `)` ã®ç·æ°ã¯ $x=i-1$ ãŸã§ã«åºãŠãã `(` ã®ç·æ°ä»¥äžã§ããïŒãŸã $x=0$ ã®ãšã A-path 㯠$\\uparrow$ ããå§ãŸãããšã¯ãªãïŒãã£ãŠäº€å·®ããããšãæ¥ããããšããªãïŒ\r\n\r\nãååæ§ããã亀差ããããšãæ¥ããããšããªããã°ïŒäžãšåæ§ã®è°è«ã§æ£ããæ¬åŒ§åã«ãªãã®ã§äžã€ç®ã®æ¡ä»¶ãæºããïŒãŸãïŒ`ã` ãš `ã` ã®éã« `()` ãšããé£ç¶éšååããªãã®ã§äºã€ç®ã®æ¡ä»¶ãæºããïŒ\r\n\r\n---\r\n\r\nããã£ãŠïŒç¹ $A$ ãã ç¹ $X$ ãŸã§ $\\to,\\uparrow$ ã ãã§è¡ãïŒç¹ $B$ ãã ç¹ $Y$ ãŸã§ $\\to,\\uparrow$ ã ãã§è¡ãçµã®ç·æ° $({}\\_{2n}\\mathrm{C}\\_{n})^2$ ããïŒA-path ãš B-path ã亀差ãŸãã¯æ¥ããçµã®ç·æ°ãåŒãã°ããïŒ\r\n\r\n---\r\n\r\n**è£é¡ïŒ**A-path ãš B-path ã®çµã§ïŒäº€å·®ãŸãã¯æ¥ãããã®ã®ç·æ°ã¯æ¬¡ã«çããïŒ\r\n$$\\dbinom{2n+1}{n}\\dbinom{2n-1}{n}$$\r\n\r\n**蚌æïŒ** \r\nã$\\to$ ãš $\\uparrow$ ãç¹ããŠåºæ¥ãç¢å°åã®ããšãåã« path ãšåŒã¶ããšã«ããïŒ\r\nA-path ãš B-path ãåããŠæ¥ããç¹ä»¥éã® path ã A-path ãš B-path ã§äº€æãã path ãèããïŒ\r\nãããšå§ç¹ãç¹ $A$ ã§çµç¹ãç¹ $Y$ ãšãªã pathãšïŒå§ç¹ãç¹ $B$ ã§çµç¹ãç¹ $X$ ãšãªã path ãæ°ããåºæ¥ãïŒå³ $2$ åç
§ïŒ\r\néã«ä»»æã®ç¹ $A$ ãšç¹ $Y$ ãçµã¶ path ãšïŒç¹ $B$ ãšç¹$X$ ãçµã¶ path ã¯å¿
ã亀差ãããæ¥ãïŒæåã«æ¥ããç¹ã§ä»¥éã® path ã亀æãããšïŒéäžã§äº€å·®ãããæ¥ãã A-path ãš B-path ã«ãªãïŒãã㧠\r\n- ç¹ $A$ ãšç¹ $Y$ ãçµã¶ path ã®ç·æ°ã¯ ${}\\_{2n+1}\\mathrm{C}\\_{n}$ åïŒ \r\n- ç¹ $B$ ãšç¹ $X$ ãçµã¶ path ã®ç·æ°ã¯ ${}\\_{2n-1}\\mathrm{C}\\_{n}$ åïŒ\r\n \r\nãªã®ã§æ±ããç·æ°ã¯è£é¡ã®éãã«ãªãïŒ\r\n\r\n\r\n\r\n---\r\n\r\nãã£ãŠäžè¬è§£ã¯\r\n$$\\dbinom{2n}{n}^2-\\dbinom{2n+1}{n}\\dbinom{2n-1}{n}=\\dfrac{1}{2n+2}\\dbinom{2n}{n}^2$$\r\nåã§ããïŒããã«æ±ããç·æ°ã¯ $n=10$ ã代å
¥ããŠ\r\n$$\\dfrac{1}{22}\\dbinom{20}{10}^2=\\textbf{1551580888}$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/3208"
},
{
"content": "ã `(` `o` `)` ã $n$ åãã€ããå ŽåãèããŸã. \r\n`o` ãš `()` ãå€æããããšã§, æ°ããã¹ã察象㯠`(`, `)` ã $2n$ åãã€çšããŠã§ããæ£ããæ¬åŒ§åã§, `()` ãäžåºŠ $n$ åå«ããã®ã§ã. \r\n\r\nãäžè¬ã«, `(` `)` ã $N$ åãã€çšããæ£ããæ¬åŒ§å㧠`()` ãäžåºŠ $K$ åãã€ãã®ã®åæ°ã¯ [ãã©ã€ãæ°](https:\\/\\/en.wikipedia.org\\/wiki\\/Narayana_number ) ãšããŠç¥ãããŠããæ°ã§, ãã®å€ã¯ $$\\frac{1}{N}\\binom{N}{K}\\binom{N}{K-1}$$\r\nã§ã. \r\n\r\nããã£ãŠä»åã®èšå®ã®ããš, æ±ããçã㯠$$\\frac{1}{2n}\\binom{2n}{n}\\binom{2n}{n-1}$$\r\nã§ã.",
"text": "ãã©ã€ãæ°ãçšãã解æ³",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/3208/547"
}
] | ãçããåæ°ã® `(` ãš `)` ãããªãæååã§ãã£ãŠïŒé£ç¶ããéšåæåå `()` ãã²ãšã€éžãã§æ¶ãããšãç¹°ãè¿ãããšã§ç©ºæååã«ã§ããæååã**æ£ããæ¬åŒ§å**ãšãã³ïŒãã®ãšãåæã«æ¶ãã `(` ãš `)` ã**察å¿ããæ¬åŒ§**ãšåŒã¶ããšã«ããŸãïŒããã¯ã©ã®æ£ããæ¬åŒ§åã«å¯ŸããŠãäžæã«å®ãŸããŸãïŒïŒ\
ã`(` ãš `)` ãš `ã` ã $10$ åãã€äžŠã¹ãŠ $30$ æåã®æåå $S$ ãäœãæ¹æ³ã®ãã¡ïŒ
- $S$ ãã `ã` ããã¹ãŠæ¶å»ãããšæ£ããæ¬åŒ§å $S^\prime$ ãåŸãããïŒ
- $S^\prime$ ã®ãã¹ãŠã®å¯Ÿå¿ããæ¬åŒ§ã¯ïŒ$S$ ã«ãããŠéã«å°ãªããšã $1$ ã€ã® `ã` ãå«ãã§ããïŒ
ã®äž¡æ¹ã®æ¡ä»¶ãæºãã䞊ã¹æ¹ã®ç·æ°ã¯å
šéšã§äœåãããŸããïŒ
<details><summary>å
·äœäŸ<\/summary>
ã以äžã¯æ¡ä»¶ãã¿ããæååã§ãïŒäžã€ã®å¯Ÿå¿ããæ¬åŒ§ãè€æ°ã® `ã` ãå«ãã§ããŠãããïŒãŸã `ã` ã¯å¿
ãããæ¬åŒ§ã®äžã«å«ãŸããŠããå¿
èŠã¯ãããŸããïŒ
- `((ã)(ã)ãã)((ã)(ã))((ã))(ã)(ãã)`ã
- `ã((ã))((ã))ãã(ã(((ã))ã)((ã)))ã`
ããŸãïŒä»¥äžã¯æ¡ä»¶ãã¿ãããªãæååã§ãïŒäžã€ç®ã¯ `ã` ãæ¶å»ããæã«æ£ããæ¬åŒ§åãšãªããïŒäºã€ç®ã«ã¯ `ã` ãéã«å«ãŸãªã察å¿ããæ¬åŒ§ãååšããŸãïŒ
- `)))))(((((ãããããããããã)))))(((((`ã
- `(ã(ã))(ã)(ã)(ã)()(ãã)(ã)(ã)(ã)`
<\/details> |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/9693 | E | OMCE005(E) | 700 | 10 | 20 | [
{
"content": "ãéè² æŽæ° $t$ ã«å¯Ÿã㊠\r\n$$f_t(x)=\\displaystyle\\sum_{i=1}^{128}\\bigg( a_i^t ~ \\prod_{j=1}^{127}\\dfrac{x-a_{i+j}}{a_i-a_{i+j}}\\bigg)$$ \r\nãšãããšïŒ$f_t(x)$ 㯠$127$ 次以äžã§ããïŒ$1\\leq k\\leq 128$ ã®æ£æŽæ° $k$ ã«å¯Ÿã㊠$f_t(a_k)=a_k^t$ ãšãªãã®ã§ïŒ$f_t(x)$ 㯠$x^t$ ã $\\displaystyle\\prod_{i=1}^{128}(x-a_i)$ ã§å²ã£ãäœãã§ããããšãåããïŒ$\\displaystyle A(x)=\\prod_{i=1}^{128}(x-a_i)=\\sum_{i=0}^{128}(-1)^iS_ix^{128-i}$ ãšããã°ïŒ$f_t$ ã $f_t(a_k)=a_k^t$ ãšãªã $127$ 次以äžã®å€é
åŒã§ããããšããïŒé©åã«ä¿æ°èª¿æŽãããŠ\r\n$$ \\begin{aligned}\r\nf_{128}(x) &= x^{128}-A(x) \\\\\\\\\r\nf_{129}(x) &= x^{129}-xA(x)-S_1A(x) \\\\\\\\\r\nf_{130}(x) &= x^{130}-x^2A(x)-S_1xA(x)+(S_2-S_1^2)A(x) \\\\\\\\\r\nf_{131}(x) &= x^{131}-x^3A(x)-S_1x^2A(x)+(S_2-S_1^2)xA(x)-(S_3-2S_1S_2+S_1^3)A(x)\r\n\\end{aligned} $$\r\nãšãªãïŒå®çŸ©ãã $A(0)=a_1a_2\\dots a_{128}$ ãã\r\n$$ f_t(0)=\\displaystyle\\sum_{i=1}^{128}\\bigg( a_i^t\\prod_{j=1}^{127}\\dfrac{-a_{i+j}}{a_i-a_{i+j}}\\bigg)=\\displaystyle\\sum_{i=1}^{128}\\bigg( a_i^{t-1}A(0)\\prod_{j=1}^{127}\\dfrac{-1}{a_i-a_{i+j}}\\bigg)$$\r\nãšãªãã®ã§ïŒ$f_{131}(0)$ ãèããã°æ±ããå€ã¯ $S_3-2S_1S_2+S_1^3$ ãšäžèŽããïŒ\r\n\r\nã$T=\\displaystyle\\sum_{i=1}^{128}a_i^2, ~ U=\\displaystyle\\sum_{i=1}^{128}a_i^3$ ãšãããšïŒæ£ $257$ è§åœ¢ã®å¯Ÿç§°æ§ãªã©ãèããã°ïŒ$256$ 以äžã®æ£æŽæ° $l$ ã«å¯Ÿã㊠$\\displaystyle\\sum_{i=1}^{128}\\cos {\\dfrac {2li}{257}\\pi}=-\\dfrac 12$ ãªã®ã§ $S_1=-\\dfrac 12$ ãšãªãïŒãããšïŒäºåè§ã®å
¬åŒãã $2T-128=-\\dfrac 12$ ãã $T=\\dfrac {255}4$ ãšãªãïŒäžåè§ã®å
¬åŒãã $4U-3S_1=-\\dfrac 12$ ãã $U=-\\dfrac 12$ ãšãªãïŒ$ S_2=\\dfrac{S_1^2-T}2, \\quad S_3=\\dfrac{S_1^3+2U-3S_1T}6$ ãçšããã°ïŒ\r\n$$ S_3-2S_1S_2+S_1^3=\\frac{126}{8} - \\frac{254}{8} - \\frac{1}{8}=-\\dfrac{129}{8} $$ \r\nããïŒæ±ããå€ã¯ $\\mathbf{137}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/9693"
},
{
"content": "$$f(z) = \\frac{ z^{130} }{ \\displaystyle \\prod_{i=1}^{128} (z-a_i) }$$\r\n\r\nãšãããšïŒæ¥µ $z = a_i$ ã«ãããçæ°ãæ±ãããåã®åé
ã«äžèŽããïŒæçé¢æ°ã® $\\mathbb{C} \\cup \\\\{ \\infty \\\\} $ ã«ãããçæ°ã®ç·å㯠$0$ ãªã®ã§ïŒä»¥äžã®ããã«å€åœ¢ã§ããïŒ\r\n\r\n$$\\begin{aligned}\r\n\\sum_{i=1}^{128} \\frac{ {a_i}^{130} }{ \\displaystyle \\prod_{j=1}^{127} (a_i-a_{i+j}) } & = \\sum_{i=1}^{128} \\mathrm{Res} (f,a_i) \\\\\\\\\r\n& = - \\mathrm{Res} (f,\\infty) \\\\\\\\\r\n& = \\mathrm{Res} \\Biggl( z^{-4} \\prod_{i=1}^{128}(1-a_i z)^{-1} ,0\\Biggr) \\\\\\\\\r\n& = \\left[z^3\\right] \\prod_{i=1}^{128} (1-a_i z)^{-1}\r\n\\end{aligned}$$\r\n\r\nåç¹ãŸãã㧠$(1 - a_i z)^{-1} = \\displaystyle\\sum_{n=0}^{\\infty} a_i^n z^n$ ã§ããããšãçšãããšïŒå
¬åŒè§£èª¬ã® $S_3 - 2S_1S_2 + S_1^3$ ãåŸãããïŒ",
"text": "çæ°ã«ããç«åŒ",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/9693/555"
},
{
"content": "ãŸã以äžã瀺ã. $$\\sum_{i=1}^{128}\\frac{a_i^{127+k}}{\\displaystyle\\prod_{j=1}^{127}(a_i-a_{i+j})}=[x^k]\\frac{1}{(1-a_1x)(1-a_2x)\\cdots(1-a_{128}x)}$$\r\n\r\n---\r\n\r\n**蚌æ**.\r\n$$\\frac{1}{(1-a_1x)(1-a_2x)\\cdots(1-a_{128}x)}$$ ãéšååæ°å解ãããšä»¥äžã®ããã«ãªã. \r\n$$\\frac{1}{(1-a_1x)(1-a_2x)\\cdots(1-a_{128}x)}=\\sum_{i=1}^{128}\\frac{1}{\\displaystyle\\prod_{j=1}^{127}(1-\\frac{a_{i+j}}{a_i})}\\frac{1}{(1-a_ix)}$$\r\n<details> <summary> 蚌æ <\\/summary>\r\nå³èŸºã®åæ¯(ã®å€é
åŒéšå)ãæã£ã以äžã®åŒã¯ $127$ 次åŒã§ãã,\r\n$$f(x)=\\sum_{i=1}^{128}\\frac{1}{\\displaystyle\\prod_{j=1}^{127}(1-\\frac{a_{i+j}}{a_i})}\\prod_{j=1}^{127}(1-a_{i+j}x)$$\r\n$x=\\frac{1}{a_k}\\quad(k=1,2,\\dots,128)$ ã代å
¥ãããš, $f(\\frac{1}{a_k})=1$ ãšãªã, $127$ 次åŒãçžç°ãªã $128$ åã®å€ã§ $f(x)=1$ ãæºããããæçåŒãšã㊠$$f(x)=1$$ ãæç«ã, äžã®éšååæ°å解ãæ£ãããšèšãã.\r\n <\\/details>\r\nãã£ãŠ\r\n$$\r\n\\begin{aligned}[x^k]\\frac{1}{(1-a_1x)(1-a_2x)\\cdots(1-a_{128}x)}&=\\sum_{i=1}^{128}\\frac{1}{\\displaystyle\\prod_{j=1}^{127}(1-\\frac{a_{i+j}}{a_i})}[x^k]\\frac{1}{(1-a_ix)}\\\\\\\\\r\n&=\\sum_{i=1}^{128}\\frac{a_i^k}{\\displaystyle\\prod_{j=1}^{127}(1-\\frac{a_{i+j}}{a_i})}\\\\\\\\\r\n&=\\sum_{i=1}^{128}\\frac{a_i^{127+k}}{\\displaystyle\\prod_{j=1}^{127}(a_i-a_{i+j})}\r\n\\end{aligned}$$\r\n\r\n---\r\n $F(x)=(1-a_1x)(1-a_2x)\\cdots(1-a_{128}x)$ ãæ±ããã.\r\n\r\nããã§, å€é
åŒ \r\n$$f(x)=(x+\\sqrt{x^2-1})^{257}+(x-\\sqrt{x^2-1})^{257}$$\r\n ã«ã€ããŠ, $f(\\cos{\\theta})=2\\cos(257\\theta)$ ãæç«ãããã, $f(x)-2$ ã¯\r\n $\\prod_{k=0}^{256}(x-\\cos {\\frac{2k\\pi}{257}})$ ã®å®æ°åã§ãã. ãã£ãŠ, \r\n$$g(x)=x^{257}(f(1\\/x)-2)=(1+\\sqrt{1-x^2})^{257}+(1-\\sqrt{1-x^2})^{257}-2x^{257}$$\r\n 㯠$\\prod_{k=0}^{256}(1-\\cos {\\frac{2k\\pi}{257}}x)=(1-x)\\\\{F(x)\\\\}^2$ ã®å®æ°åãšãªã. $\\frac{g(x)}{1-x}$ ãå±éãããšãã® $x^3$ ãŸã§ã®ä¿æ°ã調ã¹ã.\r\n$$\\begin{aligned}\r\n\\frac{g(x)}{1-x}\r\n&=2\\frac{\\sum_{k=0}^{128}\\binom{257}{2k}(1-x^2)^k-x^{257}}{1-x}\\\\\\\\\r\n&=2\\frac{\\sum_{k=1}^{128}\\binom{257}{2k}(1-x^2)^k+1-x^{257}}{1-x}\\\\\\\\\r\n&=2\\Big(\\Big(\\sum_{k=1}^{128}\\binom{257}{2k}(1+x)(1-x)^{k-1}\\Big)+1+x+x^2+\\dots+x^{256}\\Big)\\\\\\\\\r\n&=2\\Big(\\Big(1+\\sum_{k=1}^{128}\\binom{257}{2k}\\Big)+\\Big(1+\\sum_{k=1}^{128}\\binom{257}{2k}\\Big)x+\\Big(1+\\sum_{k=1}^{128}-(k-1)\\binom{257}{2k}\\Big)x^2+\\Big(1+\\sum_{k=1}^{128}-(k-1)\\binom{257}{2k}\\Big)x^3+(4次以äžã®å€é
åŒ)\\Big)\\\\\\\\\r\n&=2\\Big(\\Big(\\sum_{k=0}^{128}\\binom{257}{2k}\\Big)+\\Big(\\sum_{k=0}^{128}\\binom{257}{2k}\\Big)x+\\Big(\\sum_{k=0}^{128}-(k-1)\\binom{257}{2k}\\Big)x^2+\\Big(\\sum_{k=0}^{128}-(k-1)\\binom{257}{2k}\\Big)x^3+(4次以äžã®å€é
åŒ)\\Big)\\\\\\\\\r\n&=2(2^{256}+2^{256}x+(2^{256}-257\\times 2^{254})x^2+(2^{256}-257\\times 2^{254})x^3+(4次以äžã®å€é
åŒ))\\\\\\\\\r\n&=2^{257}\\Big(1+x+\\Big(1-\\frac{257}{4}\\Big)x^2+\\Big(1-\\frac{257}{4}\\Big)x^3+\\dots\\Big)\r\n\\end{aligned}\r\n$$\r\nãã£ãŠ, $\\\\{F(x)\\\\}^2=1+x+(1-\\frac{\r\n257}{4})x^2+(1-\\frac{\r\n257}{4})x^3+\\dots$ ã§ãã,\r\n$\\\\{F(x)\\\\}=1+c_1x+c_2x^2+c_3x^3+\\dots$ ãšãããš, $(c_1,c_2,c_3)=(\\frac{1}{2},-\\frac{127}{4},\\frac{253}{4})$ ãšãªã,æ±ããå€ã¯\r\n$$\r\n\\begin{aligned}\r\n[x^3]\\frac{1}{(1-a_1x)(1-a_2x)\\cdots(1-a_{128}x)}&=[x^3]\\frac{1}{1+c_1x+c_2x^2+cx^3+(4次以äžã®å€é
åŒ)}\\\\\\\\\r\n&=-c_1^3+2c_1c_2-c_3\\\\\\\\\r\n\\end{aligned}\r\n$$\r\nãšãªã.",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/9693/624"
}
] | ã$1\leq k \leq 128$ ãæºããæŽæ° $k$ ã«å¯Ÿã㊠$a_k=\cos{\dfrac {2k}{257}\pi}$ ãšãïŒ$129\leq k \leq 256$ ãæºããæŽæ° $k$ ã«å¯Ÿã㊠$a_k=a_{k-128}$ ãšããŸãïŒãã®ãšãïŒ
$$\large\sum_{i=1}^{128} \normalsize\dfrac{a_i^{130}}{\small\displaystyle\prod_{j=1}^{127}\normalsize(a_i-a_{i+j})}$$
ã®å€ã¯äºãã«çŽ ãªæ£ã®æŽæ° $m,n$ ãçšã㊠$-\dfrac mn$ ãšè¡šãããã®ã§ïŒ$m+n$ ã®å€ãæ±ããŠäžããïŒ |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/7423 | F | OMCE005(F) | 700 | 6 | 12 | [
{
"content": "ãäžè¬ã«ïŒ$AP = x, DP = y$ ã®å ŽåãèããïŒ\\\r\nãçŽç· $DP$ ãšäžè§åœ¢ $ABC$ ã®å
æ¥åã®äº€ç¹ã®ãã¡ $D$ ã§ãªãæ¹ã $Q$ ãšãïŒäžè§åœ¢ $ABC$ ã®å
æ¥åã® $Q$ ã§ã®æ¥ç·ãšèŸº $AB, AC$ ã®äº€ç¹ã $R, S$ ãšããïŒ\\\r\n$$\\begin{aligned}\r\n\\angle BRS\r\n&= 180^\\circ - 2\\angle FQR\\\\\\\\\r\n&= 180^\\circ - 2\\angle FDQ\\\\\\\\\r\n&= 180^\\circ - 2(90^\\circ - \\angle EFD)\\\\\\\\\r\n&= 2\\angle CED\\\\\\\\\r\n&= 180^\\circ - \\angle BCS\r\n\\end{aligned}$$\r\nããïŒåè§åœ¢ $BCSR$ ã¯åã«å
æ¥ããïŒãã®åã®äžå¿ã $M$ ãšãïŒçŽç· $BC$ ãš $EF$ ã®äº€ç¹ã $X$ ãšããïŒå
è§åœ¢ $BCESRF, BDCSQR$ ãžã®Brianchonã®å®çãã $4$ çŽç· $BS, CR, DQ, EF$ 㯠$P$ ã§äº€ããã®ã§ïŒBrocardã®å®çããïŒçŽç· $AP$ ãš $XM$ ã¯çŽäº€ããïŒããã§ïŒçŽç· $AP$ ã¯èŸº $BC$ ãšãçŽäº€ããã®ã§ïŒ$M$ ã¯èŸº $BC$ ã®äžç¹ã§ããïŒåŸã£ãŠïŒ$\\angle BRC = \\angle BSC = 90^\\circ$ ã§ããããïŒ$P$ ã¯äžè§åœ¢ $ABC$ ã®åå¿ã§ããïŒ\\\r\nãäžè§åœ¢ $ABC$ ã®å
å¿ã $I$ ãšãïŒçŽç· $AP$ ãš $BC$ ã®äº€ç¹ã $H$ ãšããïŒ\r\n$$\\angle IBR + \\angle ICS = 180^\\circ - (\\angle IRS + \\angle ISR) = \\angle RIS$$\r\nã§ããããïŒæ¥åŒŠå®çããäžè§åœ¢ $BIR$ ã®å€æ¥åãšäžè§åœ¢ $CIS$ ã®å€æ¥åã¯æ¥ããïŒããã§ïŒãã®äºã€ã®åãšåè§åœ¢ $BCSR$ ã®å€æ¥åã®æ ¹å¿ã¯ $A$ ã§ããããïŒ\r\n$$AI^2 = AB \\times AR = AH\\times AP$$\r\nã§ããïŒåŸã£ãŠïŒåè§åœ¢ $APDI$ ãå¹³è¡å蟺圢ã§ããããšãšäœµããŠïŒ$AH = \\dfrac{y^2}{x}$ ã§ããïŒãŸãïŒäžå¹³æ¹ã®å®çãã\r\n$$DH = \\sqrt{y^2 - (AH - x)^2} = \\frac{\\sqrt{3x^2y^2 - x^4 - y^4}}{x}$$\r\nã§ããïŒ\\\r\nãäžè§åœ¢ $ABC$ ã®å€å¿ã $O$ ãšãïŒ$A$ ã«å¯Ÿããåæ¥åãšèŸº $BC$ ã®æ¥ç¹ã $T$ïŒç·å $AH$ ã®äžç¹ã $N$ ãšããïŒç·å $DT$ ã®äžç¹ã¯ $M$ ã§ããïŒ$MO = 0.5x$ ã§ããããïŒ$3$ ç¹ $I,O,T$ ã¯åäžçŽç·äžã«ããïŒãŸãïŒ$3$ ç¹ $I,N,T$ ãåäžçŽç·äžã«ããã®ã§ïŒ$3$ ç¹ $I, N, O$ ã¯åäžçŽç·äžã«ããããšãåããïŒ\r\n<details><summary>ç·å $DT$ ã®äžç¹ã $M$ ã§ããããšã®èšŒæ<\\/summary>\r\nãäžè§åœ¢ $ABC$ ã® $A$ ã«å¯Ÿããåå¿ã $J$ ãšãïŒäžè§åœ¢ $ABC$ ã®å€æ¥åã® $A$ ãå«ãŸãªã匧 $BC$ ã®äžç¹ã $K$ ãšããïŒ\\\r\nã$\\angle IBJ = \\angle ICJ = 90^\\circ$ ã§ããããïŒ$4$ ç¹ $B,C,I,J$ ã¯ç·å $IJ$ ãçŽåŸãšããåäžã«ååšããïŒãŸãïŒ\r\n$$\\angle IBK = \\angle IBC + \\angle CBK = \\angle \\angle ABI + \\angle CAK = \\angle ABI + \\angle IAB = \\angle BIK$$\r\nãã $BK = IK$ ã§ããïŒåæ§ã«ã㊠$CK = IK$ ãªã®ã§ïŒ$I$ ã¯äžè§åœ¢ $BCI$ ã®å€å¿ã§ããïŒåŸã£ãŠïŒ$K$ ã¯ç·å $IJ$ ã®äžç¹ã§ããïŒããã§ïŒ$D,M,T$ 㯠$I,K,J$ ãçŽç· $BC$ ã«æ£å°åœ±ããç¹ã§ããã®ã§ïŒ$M$ ã¯ç·å $DT$ ã®äžç¹ã§ããïŒ\r\n<\\/details>\r\n<details><summary>$3$ ç¹ $I,N,T$ ãåäžçŽç·äžã«ããããšã®èšŒæ<\\/summary>\r\nã$I$ ã«é¢ã㊠$D$ ãšå¯Ÿç§°ãªç¹ã $D^\\prime$ ãšããïŒ$D^\\prime$ ã§ã®äžè§åœ¢ $ABC$ ã®å
æ¥åã®æ¥ç·ãš $T$ ã§ã®äžè§åœ¢ $ABC$ ã® $A$ ã«å¯Ÿããåæ¥åã®æ¥ç·ã¯å¹³è¡ã§ããããïŒ$A$ ãäžå¿ãšããäžè§åœ¢ $ABC$ ã®å
æ¥åãš $A$ ã«å¯Ÿããåæ¥åã®çžäŒŒã«ãã㊠$D^\\prime$ ãš $T$ ã¯å¯Ÿå¿ããïŒåŸã£ãŠïŒ$A,D^\\prime, T$ ã¯åäžçŽç·äžã«ããïŒãŸãïŒ$H,D,T$ ãåäžçŽç·äžã«ããïŒçŽç· $AH$ ãš $D^\\prime D$ ã¯å¹³è¡ã§ããããïŒç·å $AH$ ã®äžç¹ $N$ ãšç·å $D^\\prime D$ ã®äžç¹ $I$ ãçµã¶çŽç·ã¯ $T$ ãéãïŒ\r\n<\\/details>\r\n\r\nä»ïŒäžå¹³æ¹ã®å®çãã\r\n$$IN^2 = DH^2 + (AN - ID)^2 = \\frac{8x^2y^2 - 3y^4}{4x^2}$$\r\n\r\nã§ããããïŒ$\\angle NAI = \\angle OAI$ ãã次ã®è£é¡ãçšããããšã§\r\n$$AO = \\frac{AN\\times AI^2}{AN^2 - IN^2} = \\frac{xy^2}{2y^2 - 4x^2}$$\r\nãåŸãïŒ\r\n\r\n----\r\n**è£é¡.**ãäžè§åœ¢ $XYZ$ ã®èŸº $YZ$ äžã« $W$ ã $\\angle YXW = \\angle ZXW$ ãæºããããã«åã£ããšãïŒæ¬¡ãæç«ããïŒ$$XZ = \\frac{XY\\times WX^2}{XY^2 - WY^2}$$\r\n**蚌æ.**ãäžè§åœ¢ $XYZ$ ã®å€æ¥åãš $WX$ ã®äº€ç¹ã $V$ ãšããïŒ\r\n$$\\angle XVZ = \\angle XYW,\\quad \\angle WXY = \\angle VXZ$$\r\nã§ããããïŒäžè§åœ¢ $WXY$ ãš $ZXV$ ã¯çžäŒŒã§ããïŒåŸã£ãŠïŒ\r\n$$XY\\times XZ = WX\\times VX$$\r\nã§ããïŒãŸãïŒæ¹ã¹ãã®å®çãã $VW\\times WX = WY\\times WZ$ ã§ããããïŒ\r\n$$WX^2 = WX\\times VX - WX\\times VW = XY\\times XZ - WY\\times WZ$$\r\nã§ããïŒããã« $WZ = \\dfrac{WY\\times XZ}{XY}$ ã代å
¥ããŠå€åœ¢ããã°ïŒææã®åŒãåŸãïŒ\r\n----\r\n\r\nåŸã£ãŠïŒ\r\n$$BC = 2BM = 2\\sqrt{AO^2 - MO^2} = \\frac{2x^2\\sqrt{y^2 - x^2}}{y^2 - 2x^2}$$\r\nã§ããããïŒæ±ããé¢ç©ã¯\r\n$$\\frac{1}{2}\\times AH\\times BC = \\frac{xy^2\\sqrt{y^2-x^2}}{y^2-2x^2}$$\r\nã§ããïŒ$(x,y)=(321,500)$ ã代å
¥ããŠèšç®ããã°ïŒè§£çãã¹ãå€ã¯ $\\bf{40293918}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/7423"
},
{
"content": "ã$AP=x, DP=y$ ãšããïŒäžè§åœ¢ $ABC$ ã® $\\angle A$ å
ã®åå¿ïŒåæ¥åã $I_A, \\omega_A$ ãšã $\\omega_A$ ãšçŽç· $BC, CA, AB$ ã®æ¥ç¹ãé ã« $D_A, E_A, F_A$ ãšããïŒçŽç· $AP$ ãšçŽç· $BC, E_AF_A$ ã®äº€ç¹ã $H, Q$ ãšãïŒç¹ $I$ ããçŽç· $AP$ ã«äžãããåç·ã®è¶³ã $J$ ãšããïŒ\\\r\nãåè§åœ¢ $APDI$ ã¯å¹³è¡å蟺圢ãªã®ã§ $AP=ID=JH$ ã§ããïŒ$A, E, I, J, F$ ã®å
±åãã\r\n$$AP\\times AJ=AF^2=AI^2-IF^2=AI^2-AP\\times JH \\rArr AI^2=AP\\times AH$$\r\nããããïŒ$A$ ãäžå¿ãšããçžäŒŒæ¡å€§ãªã©ãã $HD:HD_A=ID:I_AD_A$ ãã $\\angle IHA=\\angle I_AHQ$ ã§ããïŒ$AP:AQ=AI:AI_A$ ãã $AH\\times AQ=AI\\times AI_A$ ãã $I, I_A, Q, H$ ãå
±åãšãªãã®ã§ïŒäžèšãšåãã㊠$QI=QI_A$ ãšãªãïŒãããã£ãŠåè§åœ¢ $IE_AI_AF_A$ ã¯ã²ã圢ãšãªã $IF_A=I_AF_A$ ããã³ $F_A, I, E$ ãåäžçŽç·äžã§ããããšãã $\\cos A=\\dfrac {IF}{IF_A}=\\dfrac {IF}{I_AF_A}=\\dfrac {AI}{AI_A}$ ãšãªãïŒãã£ãŠæ±ããäžè§åœ¢ $ABC$ ã®é¢ç©ã¯\r\n$$\\dfrac 12AB\\times AC\\sin A=\\dfrac 12AI\\times AI_A\\sin A=\\dfrac 12AI^2\\tan A$$\r\nããã㧠$\\sin \\dfrac A2=\\dfrac xy$ ãã $\\tan A=\\dfrac {2\\sin \\frac A2 \\cos \\frac A2}{1-2\\sin ^2\\frac A2}=\\dfrac {2x\\sqrt {y^2-x^2}}{y^2-2x^2}$ ãªã®ã§æ±ããå€ã¯ $\\dfrac {xy^2\\sqrt {y^2-x^2}}{y^2-2x^2}$ ã§ããïŒ",
"text": "åå¿ãçšãã解æ³",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/7423/546"
},
{
"content": "$EF$ ã®äžç¹ã $N$ ãšããïŒ$x=500,y=321$ ãšãããšåè§åœ¢ $APDI$ ã¯å¹³è¡å蟺圢ãªã®ã§ $AI=x,EI=DI=y$ïŒãã®ãã $IN=\\frac{y^2}{x}$ ã§ïŒ$AN=x-\\frac{y^2}{x}=\\frac{x^2-y^2}{x}$ïŒ\\\r\n$AP$ ãš $BC$ ã®äº€ç¹ã $H$ ãšãããšäžè§åœ¢ $ANP,PHD$ ã¯çžäŒŒãªã®ã§ $PH=AN\\cdot \\frac{PD}{AP}=\\frac{y^2-x^2}{y}$ïŒãã£ãŠ $AH=y+PH=\\frac{x^2}{y}$ïŒ\\\r\näžè§åœ¢ $ABC$ ã®é¢ç©ã $S$ ãšãïŒèŸº $BC,CA,AB$ ã®é·ãã $a,b,c$ ãšãããš $aAH=(a+b+c)DI=2S$ ãšãªãã®ã§ $a:a+b+c=DI:AH=y^2:x^2$ïŒæ
ã« $a:\\frac{b+c-a}{2}=y^2:\\frac{x^2-2y^2}{2}$ ã§ããïŒ\\\r\nãã㧠$AE=\\frac{b+c-a}{2}$ ã§ïŒ$AE=\\sqrt{AI^2-EI^2}=\\sqrt{x^2-y^2}$ ã§ããããïŒ$a=\\sqrt{x^2-y^2}\\frac{2y^2}{x^2-2y^2}$ïŒ\\\r\nãã£ãŠ $S=aAH\\frac{1}{2}=\\sqrt{x^2-y^2}\\frac{2y^2}{x^2-2y^2}\\cdot \\frac{x^2}{2y}=\\sqrt{x^2-y^2}\\frac{y}{x^2-2y^2}x^2$\r\nåŸã¯ä»£å
¥ããã ãïŒ\\\r\n\\\r\nè¿œèšãæ¬è§£ãš $x,y$ ãéã«ããŠããŸã£ãããã",
"text": "åšãå©çšããæ¹æ³",
"url": "https://onlinemathcontest.com/contests/omce005/editorial/7423/554"
}
] | ã$AB\neq AC$ ãªãäžè§åœ¢ $ABC$ ã®å
æ¥åãšèŸº $BC, CA, AB$ ã®æ¥ç¹ããããã $D,E,F$ ãšãïŒ$D$ ããçŽç· $EF$ ã«äžãããåç·ã®è¶³ã $P$ ãšãããšïŒ$AP \perp BC$ ãšãªããŸããïŒããã«ïŒ
$$AP = 321,\quad DP = 500$$
ã§ãããšãïŒäžè§åœ¢ $ABC$ ã®é¢ç©ãæ±ããŠãã ããïŒãã ãïŒæ±ããçãã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãšå¹³æ¹å åãæããªãæ£ã®æŽæ° $c$ ãçšã㊠$\dfrac{b\sqrt c}{a}$ ãšè¡šãããã®ã§ïŒ$a+b+c$ ã解çããŠãã ããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/7069 | A | OMC223(A) | 300 | 155 | 252 | [
{
"content": "ã$30^6 = 2^6 \\cdot 3^6 \\cdot 5^6$ ã $3$ ã€ã®æ£æŽæ°ã®ç©ã§è¡šãæ¹æ³ïŒããç®ã®é åºã¯åºå¥ããªãïŒã®ãã¡ïŒ$3$ æ°ãçžç°ãªããã®ã®åæ°ã $M$ïŒ$2$ æ°ã®ã¿ãçãããã®ã®åæ°ã $N$ ãšããïŒãã®ãšã $abc = 30^6$ ãªãæ£æŽæ°ã®çµ $(a, b, c)$ ã®ç·æ°ã¯ïŒæåã®å¯Ÿç§°æ§ã $a = b = c$ ã®ã±ãŒã¹ãèæ
®ãããš $6M + 3N + 1$ ãšè¡šãããšãã§ããïŒãã®ãããªçµã®åæ°ãå®éã«èšç®ãããšïŒ$a, b, c$ ã®çŽ å æ°å解ã«å²ãæ¯ããã $2, 3, 5$ ã®éè€åºŠã®æ±ºãæ¹ãåã
${}\\_{8}\\mathrm{C}\\_{2} = 28$ éããã€ããããïŒ$28^3$ ã§ããïŒãã£ãŠïŒ\r\n$$6M + 3N + 1 = 28^3 \\implies 2M + N = 7317$$\r\nãåŸãããïŒäžæ¹ã§ïŒç©ã $30^6$ ã§ãã $2$ æ°ã®ã¿ãçãããªããã㪠$3$ ã€ã®æ£æŽæ°ã®çµã¿åãã㯠$30^2$ 以å€ã® $30^3$ ã®æ£ã®çŽæ° $d$ ãçšã㊠$(d, d, 30^6\\/d^2)$ ãšè¡šãããšãã§ãïŒ$30^3$ ã®æ£ã®çŽæ°ã¯å
šéšã§ $64$ åãªã®ã§ïŒãããã $N = 64 - 1 = 63$ ãåããïŒå
ã®åŒãšåãã㊠$M = 3627$ ãåããïŒ\\\r\nãããã§ïŒåé¡ã®æ¡ä»¶ãæºããçµ $(x, y, z)$ ã¯ïŒä»¥äžã® $2$ ã€ã«åé¡ããããšãã§ããïŒ\r\n- $|x|, |y|, |z|$ ãçžç°ãªãæ£æŽæ°ãšãªããã®ïŒ\r\n- çžç°ãªãæ£æŽæ° $p, q$ ã«ãã£ãŠïŒ$x, y, z$ ã $p, -p, -q$ ãå°ããé ã«äžŠã¹ããã®ãšãªããã®ïŒ\r\n\r\n$1$ çªç®ã«åé¡ããããã®ã¯ïŒç¬Šå·ã®å²ãæ¯ãã $4$ ãã¿ãŒã³ããããšã«æ³šæãããšå
šéšã§ $4M$ åããããšãåããïŒ$2$ çªç®ã«åé¡ããããã®ã¯å
šéšã§ $N$ åããããšãåããïŒããã«æ±ããåæ°ã¯\r\n$$4M + N = \\mathbf{14571}$$\r\n\r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/7069"
},
{
"content": "ã$x,y,z$ ãçžç°ãªããããªçµã®ç·æ°ã $6$ ã§å²ã£ãå€ãæ±ããå€ã«çããïŒä»£ããã«ïŒã$x,y,z$ ã®ãã¡ïŒå°ãªããšã $2$ ã€ã¯çããããã㪠$(x,y,z)$ ã®ç·æ° $S$ ãæ°ããïŒ\r\n\r\nã$x=y, ~ y=z, ~ z=x$ ãªã $(x,y,z)$ ã®éåããããã $A,B,C$ ãšããïŒãããšå
é€åçïŒããã³ $A,B,C$ ã«å¯Ÿãã察称æ§ãã\r\n$$S = |A \\cup B \\cup C| = 3|A| - 3|A \\cap B| + |A \\cap B \\cap C|$$\r\nãåããïŒãŸã $A \\cap B = A \\cap B \\cap C$ ã§ããããïŒ\r\n$$S = 3|A| - 2|A \\cap B|$$\r\nã§ããïŒ\\\r\nã$|A|$ ã«ã€ããŠïŒ$30^6$ ã®å¹³æ¹æ°ã®æ£ã®çŽæ°ã¯ $(3+1)^3 = 64$ åããïŒç¬Šå·ãèæ
®ããã°ãããã $2$ éãããããïŒ$|A| = 128$ ãåããïŒ\\\r\nã$|A \\cap B|$ ã«ã€ããŠïŒãã㯠$x=y=z=30^2$ ã«éãããã®ã§ $|A \\cap B| = 1$ïŒ\r\n\r\nã以äžããïŒ$S = 382$ ãªã®ã§ïŒæ±ããå€ã¯\r\n$$\\frac{4Ã28^3-S}{6} = \\mathbf{14571}.$$",
"text": "å
é€åç",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/7069/543"
}
] | ã$3$ ã€ã®ïŒæ£ãšã¯éããªãïŒæŽæ°ã®çµ $(x, y, z)$ ã§ãã£ãŠïŒ$x \lt y \lt z$ ã〠$xyz = 30^6$ ãã¿ãããã®ã¯ããã€ãããŸããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/8509 | B | OMC223(B) | 300 | 136 | 206 | [
{
"content": "ãäžè¬ã« $n\\geq 3$ ã«å¯ŸããŠæ£ $2n$ è§åœ¢ $P_1P_2 \\ldots P_{2n}$ ãèãïŒãã®å€æ¥åã $\\Omega$ ãšããïŒä»¥äžãæºãããããªçµ $(i_1, i_2, i_3, i_4, i_5)$ ã®åæ°ãæ±ããã°ããïŒ\r\n- äºè§åœ¢ $P_{i_1}P_{i_2}P_{i_3}P_{i_4}P_{i_5}$ ã® $5$ ã€ã®å¯Ÿè§ç·ã®ãã¡å°ãªããšã $1$ ã€ã¯å $\\Omega$ ã®çŽåŸã§ããïŒ\r\n\r\nããã§çŽåŸãšãªãåŸã察è§ç·ã¯é«ã
$2$ ã€ã§ããããšã«æ³šæããïŒäºè§åœ¢ã®å¯Ÿè§ç·ãšãªãããçŽåŸã¯\r\n$$P_1P_{n+1}ïŒP_2P_{n+2}ïŒ\\ldotsïŒP_nP_{2n}$$\r\nã® $n$ æ¬ã§ããïŒãŸãã¯ãããã®äžãã $1$ ã€ãéžã³åºå®ãããšãã«ïŒããã察è§ç·ã«å«ããããªäºè§åœ¢ã®åæ° $A$ ãæ±ããïŒçŽåŸã®ç«¯ç¹ä»¥å€ã® $3$ ç¹ã¯ïŒçŽåŸãå¢ãšãã $2$ é åã«ãããã $1, 2$ åå«ãŸããïŒ$2$ åå«ãŸããåŽã®æ±ºãæ¹ã¯ $2$ éãããïŒåé åã«ãããé ç¹ã®äœçœ®ã®éžã³æ¹ã¯ãããã $n - 1, {}\\_{n - 1}\\mathrm{C}\\_{2}$ éãããïŒããã«\r\n$$A = 2 \\times (n - 1) \\times {}\\_{n - 1}\\mathrm{C}\\_{2} = (n - 1)^2(n - 2)$$\r\nã§ããïŒ\\\r\nã次ã«çŽåŸã®äžãã $2$ ã€ãéžã³åºå®ãããšãã«ïŒãããã察è§ç·ã«å«ããããªäºè§åœ¢ã®åæ°ãæ±ãããïŒããã¯çŽåŸã®ç«¯ç¹ãšãªã $4$ ç¹ä»¥å€ã®äžãã $1$ ç¹ãéžã¶æ¹æ³ã«çããã®ã§ïŒ$2n - 4$ éãã§ããïŒ\\\r\nã$n$ åã®çŽåŸãã¹ãŠã«å¯ŸããŠããã察è§ç·ã«å«ãäºè§åœ¢ã®åæ°ãæ°ãããšãã®åæ°ã¯ $nA$ ãšãªããïŒãã®æ°ãæ¹ã§ã¯çŽåŸã $2$ ã€å«ãäºè§åœ¢ããããã $2$ åãã€ã«ãŠã³ããããã®ã§ïŒãã®éè€ãèæ
®ãããšæ±ããåæ°ã¯\r\n$$nA - {}\\_{n}\\mathrm{C}\\_{2} (2n - 4) = n(n - 1)(n - 2)^2$$\r\nã§ããïŒ$n = 555$ ã§ãã㯠$\\mathbf{94027093230}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/8509"
},
{
"content": "ã$n=555$ ãšããïŒåé¡ã®æ¡ä»¶ãæºãããªããã㪠$(i_1,i_2,i_3,i_4,i_5)$ ã®ç·æ°ãæ°ãããïŒããã¯æ¬¡ã® $2$ ã€ã®å Žåã«åããããïŒ\r\n\r\n$(1)$ $P_{i_1},P_{i_2},P_{i_3},P_{i_4},P_{i_5}$ ã®ãã¡ã®ã©ã® $2$ ç¹ã®çµãçŽåŸã§ãªãïŒ\r\n\r\n$(2)$ $P_{i_1}P_{i_2},P_{i_2}P_{i_3},P_{i_3}P_{i_4},P_{i_4}P_{i_5},P_{i_5}P_{i_1}$ ã®ãã¡ïŒãããããçŽåŸïŒ\r\n\r\nã $\\\\{ i_1,i_2,i_3,i_4,i_5 \\\\}$ ã $1$ ä»¥äž $2n$ 以äžã®æŽæ°ããåºå¥ãªã $5$ ã€éžã¶ãã®ãšèããïŒ\r\n\r\n- $(1)$ ã«ã€ã㊠\\\r\nã$n$ æ¬ã®çŽåŸã®äžãã $5$ æ¬ãéžã³ïŒããã«ã©ã¡ãã®ç«¯ãç¹ãšããŠéžã¶ã決ããã°è¯ãïŒ\r\n\r\n- $(2)$ ã«ã€ã㊠\\\r\nã$n$ æ¬ã®çŽåŸã®äžãã $1$ æ¬éžã³ïŒãã®äž¡ç«¯ã®ç¹ã $2$ ç¹ãšããïŒæ®ãã® $3$ ç¹ã¯ïŒéžãã çŽåŸã«ãã£ãŠåãããã $2$ é åã®äžããçæ¹ãéžã³ïŒããã® $n-1$ ç¹ãã $3$ ã€éžã¹ã°è¯ãïŒ\r\n\r\nããã£ãŠæ±ããå€ã¯ïŒ\r\n$${}\\_{2n}\\mathrm{C}\\_{5} - \\underbrace{ {}\\_{n}\\mathrm{C}\\_{5}Ã2^5}\\_{(1)} - \\underbrace{ nÃ2Ã{}\\_{n-1}\\mathrm{C}\\_{3}}\\_{(2)} = \\mathbf{94027093230}$$\r\nã§ããïŒ",
"text": "äœäºè±¡ãæ°ãã",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/8509/548"
}
] | ãæ£ $1110$ è§åœ¢ $P_1P_2 \cdots P_{1110}$ ã«å¯ŸãïŒ
$$1 \leq i_1 \lt i_2 \lt i_3 \lt i_4 \lt i_5 \leq 1110$$
ãªãæŽæ°ã®çµ $(i_1, i_2, i_3, i_4, i_5)$ ã§ãã£ãŠïŒä»¥äžãã¿ãããã®ã¯å
šéšã§ããã€ãããŸããïŒ
- äºè§åœ¢ $P_{i_1}P_{i_2}P_{i_3}P_{i_4}P_{i_5}$ ã®å
è§ã®ãã¡ïŒå°ãªããšã $1$ ã€ã¯çŽè§ã§ããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/8513 | C | OMC223(C) | 300 | 116 | 150 | [
{
"content": "ãä»»æã®å®æ° $x$ ã§\r\n$$g(x) = -f(-x)$$\r\nãæãç«ã¡ïŒãŸãïŒä»»æã® $0$ ãé€ãå®æ° $x$ ã§\r\n$$h(x) = x^3f \\left (\\frac{1}{x} \\right )$$\r\nãæãç«ã€ïŒããã«åé¡ã®æ¡ä»¶\r\n$$g \\left (\\frac{11}{10} \\right ) = h(1110) = 0$$\r\nã¯\r\n$$f \\left (- \\frac{11}{10} \\right ) = f \\left (\\frac{1}{1110} \\right ) = 0$$\r\nãšåå€ã§ããïŒ$- \\dfrac{11}{10}, \\dfrac{1}{1110}$ 㯠$f(x)$ ã®æ ¹ã§ããïŒ$f(x)$ ãæ ¹ã $2$ ã€ä»¥äžãã€ããšãã $a \\neq 0$ ãåŸãïŒ$f(x)$ 㯠$3$ 次åŒã§ããïŒãã㧠$f(x)$ ã®ç¬¬äžã®æ ¹ã $\\alpha$ ãšããã°ïŒè§£ãšä¿æ°ã®é¢ä¿ãã\r\n$$- \\frac{11}{10} + \\frac{1}{1110} + \\alpha = 0$$\r\nãæãç«ã€ã®ã§ãããã $\\alpha = \\dfrac{122}{111}$ ãåŸãããïŒãã® $\\alpha$ ãæ±ããæ倧å€ã§ããïŒç¹ã«è§£çãã¹ãå€ã¯ $\\mathbf{233}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/8513"
}
] | ã$a, b$ ãå®æ°ãšãïŒå®æ°ã«å¯ŸããŠå®çŸ©ãããé¢æ° $f, g, h $ ããããã
$$
\begin{aligned}
f(x) &= ax^3 + x + b, \\\\
g(x) &= ax^3 + x - b, \\\\
h(x) &= bx^3 + x^2 + a\\\\
\end{aligned}
$$
ã§å®ããŸãïŒ
$$g \left (\frac{11}{10} \right ) = h(1110) = 0$$
ãæãç«ã€ãšãïŒ$f(x) = 0$ ãæºããæ倧ã®å®æ° $x$ ãæ±ããŠãã ããïŒãã ãæ±ããæ倧å€ã¯äºãã«çŽ ãªæ£æŽæ° $p ,q$ ã«ãã£ãŠ $\dfrac{p}{q}$ ãšè¡šãããšãã§ããã®ã§ïŒ$p + q$ ã®å€ã解çããŠäžããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/10404 | D | OMC223(D) | 400 | 72 | 134 | [
{
"content": "ãäžããããçåŒãå€åœ¢ãããš\r\n$$pm^2 = q^n(q^n + 32)(q^n - 32)$$\r\nãšãªãïŒ$q^n$ ã¯å¥æ°ããïŒ$q^n - 32, q^n, q^n + 32$ ã¯ã©ã® $2$ ã€ãéžãã§ãäºãã«çŽ ã§ããïŒ$pm^2$ ãçŽ å æ°å解ãããšãã«ã¹ããå¥æ°ãšãªãçŽ æ°ã $p$ ãã $1$ ã€ã§ããããšããïŒ$q^n - 32, q^n, q^n + 32$ ã®ãã¡ã¡ããã© $2$ ã€ãå¹³æ¹æ°ãšãªãå¿
èŠãããïŒããã§è£é¡ãäžããïŒ\r\n\r\n---\r\n**è£é¡ïŒ**\\\r\nãå¥æ°ã®å¹³æ¹æ° $A, B$ ãš $3$ 以äžã®æŽæ° $k$ ã®é㧠$A = B + 2^k$ ãæãç«ã€ãšãïŒ\r\n$$(A, B) = \\left( (2^{k-2} + 1)^2, (2^{k-2} - 1)^2 \\right)$$\r\nãããããïŒ\r\n\r\n<details><summary>è£é¡ã®èšŒæ<\\/summary>\r\nãæ£ã®å¥æ° $M, N$ ãçšã㊠$A = M^2, B = N^2$ ãšè¡šããš\r\n$$(M + N)(M - N) = A - B = 2^k$$\r\nã§ããïŒ$M + N \\gt M - N$ ã〠$(M + N) - (M - N) = 2M$ ã $4$ ã§å²ãåããªãå¶æ°ã§ããããšãã\r\n$$(M + N, M - N) = (2^{k - 1}, 2)$$\r\nãããããïŒãã£ãŠ $(M, N) = (2^{k-2} + 1, 2^{k-2} - 1)$ ãåŸããïŒäž»åŒµã¯æãç«ã€ïŒ\r\n<\\/details>\r\n\r\n---\r\n\r\nããã§ã¯ïŒ$q^n - 32, q^n, q^n + 32$ ã®äžããå¹³æ¹æ°ãšãªã $2$ æ°ãéžã¶æ¹æ³ã«ãã£ãŠïŒäžèšã®ããã«å Žååããè°è«ãããïŒ\r\n\r\n---\r\n- $q^n, q^n - 32$ ãå¹³æ¹æ°ã®ãšãïŒ \\\r\nè£é¡ãã $q^n = (2^3 + 1)^2 = 81$ ãããããïŒ$(q, n) = (3, 4)$ ãåŸãïŒããã«\r\n$$pm^2 = 49 \\times 81 \\times 113 = 113 \\times 63^2$$\r\nãæãç«ã€ã®ã§ïŒãã®ã±ãŒã¹ã§ã¯ $(p, q, m, n) = (113, 3, 63, 4)$ ãé©ããïŒ\r\n\r\n- $q^n, q^n + 32$ ãå¹³æ¹æ°ã®ãšãïŒ \\\r\nè£é¡ãã $q^n = (2^3 - 1)^2 = 49$ ãããããïŒ$(q, n) = (7, 2)$ ãåŸãïŒããã«\r\n$$pm^2 = 17 \\times 49 \\times 81 = 17 \\times 63^2$$\r\nãæãç«ã€ã®ã§ïŒãã®ã±ãŒã¹ã§ã¯ $(p, q, m, n) = (17, 7, 63, 2)$ ãé©ããïŒ\r\n\r\n- $q^n - 32, q^n + 32$ ãå¹³æ¹æ°ã®ãšãïŒ \\\r\nè£é¡ãã $q^n - 32 = (2^4 - 1)^2$ ãªã®ã§ïŒ$q^n = 257$ ããããã $(q, n) = (257, 1)$ ãåŸãïŒããã«\r\n$$pm^2 = 225 \\times 257 \\times 289 = 257 \\times 255^2$$\r\nãæãç«ã€ã®ã§ïŒãã®ã±ãŒã¹ã§ã¯ $(p, q, m, n) = (257, 257, 255, 1)$ ãé©ããïŒ\r\n\r\n---\r\n\r\nã以äžã®è°è«ããïŒæ±ããç·ç©ã¯ $17 \\times 113 \\times 257 = \\mathbf{493697}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/10404"
}
] | ãæ£æŽæ° $m, n$ ãš $3$ 以äžã®çŽ æ° $p, q$ ã
$$pm^2 + 1024q^n = q^{3n}$$
ãã¿ãããŠããŸãïŒ$p$ ã®å€ãšããŠããåŸããã®ã®**ç·ç©**ã解çããŠäžããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/10609 | E | OMC223(E) | 400 | 22 | 55 | [
{
"content": "ã$c_1, c_2, c_3, c_4, c_5$ ãåé¡ã®æ¡ä»¶ãã¿ãããšãïŒ\r\n$$\\max \\\\{c_1, c_2, c_3, c_4, c_5\\\\} \\leq 40000$$\r\nã§ããã®ã§ïŒ\r\n$$\\sum_{n = 1}^{40000} na_n = \\sum_{i = 1}^5 \\sum_{k = 1}^{c_i} k$$\r\nãšè¡šãããšãã§ããïŒãã ã $\\displaystyle \\sum_{k = 1}^0 k = 0$ ãšããïŒïŒãããã£ãŠïŒ\r\n$$\\sum_{i = 1}^5 (2i - 1)x_i = 40000 \\tag{1}$$\r\nãªãéè² æŽæ°ã®çµ $(x_1, x_2, x_3, x_4, x_5)$ ã«å¯Ÿãã\r\n$$\\sum_{i = 1}^5 \\sum_{k = 1}^{x_i} k$$\r\nã®æå°å€ãæ±ããåé¡ã«åž°çã§ããïŒ$c_1 \\geq c_2 \\geq c_3 \\geq c_4 \\geq c_5$ ãªã®ã§ïŒ$x_1, x_2, x_3, x_4, x_5$ ã倧ããæ¹ããé ã« $c_1, c_2, c_3, c_4, c_5$ ãšå¯Ÿå¿ä»ããŠïŒé©åœãªäžŠã³æ¿ã $m_1, m_2, m_3, m_4, m_5$ ãéžã¹ã°ããïŒïŒããã§ïŒ\r\n$$\r\n\\begin{aligned}\r\n\\sum_{i = 1}^5 \\sum_{k = 1}^{x_i} k &= \\sum_{i = 1}^5 \\frac{x_i(x_i + 1)}{2} = \\frac{1}{8} \\left ( \\sum_{i = 1}^5 (2x_i + 1)^2 - 5\\right ) \\\\\\\\\r\n&= \\frac{1}{8 \\times 165} \\left ( \\sum_{i = 1}^5 (2x_i + 1)^2 \\sum_{j = 1}^5 (2j - 1)^2 - 5 \\times 165\\right )\r\n\\end{aligned}\r\n$$\r\nãšè¡šããã®ã§ïŒCauchyâSchwarzã®äžçåŒãçšããããšã§åŒ $(1)$ ãã\r\n$$\r\n\\begin{aligned}\r\n\\sum_{i = 1}^5 (2x_i + 1)^2 \\sum_{j = 1}^5 (2j - 1)^2 &\\geq \\left ( \\sum_{i = 1}^5 (2i - 1)(2x_i + 1) \\right )^2 \\\\\\\\\r\n&= \\left ( 2 \\times 40000 + \\sum_{i = 1}^5 (2i - 1) \\right )^2 \\\\\\\\\r\n&= 80025^2\r\n\\end{aligned}\r\n$$\r\nãåŸããïŒçå·ãæç«ããå¿
èŠååæ¡ä»¶ã¯\r\n$$2x_1 + 1 : 2x_2 + 1 : 2x_3 + 1 : 2x_4 + 1 : 2x_5 + 1 = 1 : 3 : 5 : 7 : 9$$\r\nã§ããïŒåŒ $(1)$ ããïŒãããæãç«ã€ã®ã¯ä»¥äžã®ãšãã§ããïŒ\r\n$$(x_1, x_2, x_3, x_4, x_5) = (242, 727, 1212, 1697, 2182)$$\r\n以äžããïŒæ±ããæå°å€ã¯\r\n$$\\frac{1}{8 \\times 165}(80025^2 - 5 \\times 165) = \\mathbf{4851515}$$\r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/10609"
},
{
"content": "ãé«æ ¡ç¯å²ããã¯éžè±ããŸããïŒLagrange ã®æªå®ä¹æ°æ³ã䜿ãæ¹éã§ãïŒ\r\n\r\n---\r\n\r\nãå
¬åŒè§£èª¬åæ§ã«\r\n$$ \\tag{1} x_1+3x_2+5x_3+7x_4+9x_5=40000$$\r\nãæºããæ¡ä»¶ã®äžã§\r\n$$\\tag{2} \\dfrac{x_1(x_1+1)}{2}+\\dfrac{x_2(x_2+1)}{2}+\\dfrac{x_3(x_3+1)}{2}+\\dfrac{x_4(x_4+1)}{2}+\\dfrac{x_5(x_5+1)}{2}$$\r\nã®æå°å€ãæ±ãããïŒ\r\n$$f(x_1, x_2, x_3, x_4, x_5)=\\dfrac{x_1(x_1+1)}{2}+\\cdots+\\dfrac{x_5(x_5+1)}{2}-\\lambda (x_1+3x_2+5x_3+7x_4+9x_5-40000)$$\r\nãšçœ®ããŠïŒãã€ãã€å埮åãããŠããïŒ\r\n\r\n$$\\begin{aligned}\r\n\\dfrac{\\partial f}{\\partial x_1} &=x_1+\\dfrac{1}{2}-\\lambda=0 \\\\\\\\\r\n\\dfrac{\\partial f}{\\partial x_2} &=x_2+\\dfrac{1}{2}-3\\lambda=0 \\\\\\\\\r\n\\vdots\\\\\\\\\r\n\\dfrac{\\partial f}{\\partial x_5} &=x_5+\\dfrac{1}{2}-9\\lambda=0\r\n\\end{aligned}$$\r\nã$x_1=\\lambda-\\dfrac{1}{2}$ çãšããŠåŒ $(1)$ ã«ä»£å
¥ããã°ïŒ$\\lambda=\\dfrac{485}{2}$ ãåŸãïŒ\\\r\nããããã $(x_1,x_2x_3,x_4,x_5)=(2182,1697,1212,727,242)$ ãåŸãã®ã§ïŒåŒ $(2)$ ã«ä»£å
¥ããã°çããåŸãïŒ\r\n\r\n---\r\n\r\nããªãïŒLagrange ã®æªå®ä¹æ°æ³ã¯æ¥µå€ïŒæ倧å€ïŒã®ååšãä¿èšŒãããã®ã§ã¯ãªãã®ã§ïŒå³å¯ã«ã¯ãã®ç¹ã«ã¯æ³šæãå¿
èŠã§ãïŒOMCã®ã«ãŒã«äžïŒãã®ãããªããšã¯èããå¿
èŠããªãã®ã§ããâŠïŒïŒè©³ããã¯é©åœãªåèæžçã§ç¢ºèªããŠãã ããïŒ\\\r\nãé¡é¡ãšããŠïŒ[OMC112(C)](https:\\/\\/onlinemathcontest.com\\/contests\\/omc112\\/tasks\\/3393)ïŒ[OMC132(E)](https:\\/\\/onlinemathcontest.com\\/contests\\/omc132\\/tasks\\/4656)ïŒ[OMC217(F)](https:\\/\\/onlinemathcontest.com\\/contests\\/omc217\\/tasks\\/9293)ãæããŠãããŸãïŒ",
"text": "Lagrange ã®æªå®ä¹æ°æ³",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/10609/549"
}
] | $$5 \geq a_1 \geq a_2 \geq \cdots \geq a_{40000} \geq 0$$
ãªãæŽæ°ã®çµ $(a_1, ..., a_{40000})$ ãããïŒéè² æŽæ° $c_1, c_2, c_3, c_4, c_5$ ã次ã®ããã«å®ããŸãïŒ
- å $i \in \\{1, 2, 3, 4, 5\\}$ ã«å¯ŸãïŒ$1 \leq n \leq 40000$ ãªãæŽæ° $n$ ã®ãã¡ $a_n \geq i$ ãã¿ãããã®ã®åæ°ã $c_i$ ãšããïŒ
ãããš $1, 2, 3, 4, 5$ ã®äžŠã¹æ¿ã $m_1, m_2, m_3, m_4, m_5$ ã§ãã£ãŠ
$$c_{m_1} + 3c_{m_2} + 5c_{m_3} + 7c_{m_4} + 9c_{m_5} = 40000$$
ãã¿ãããã®ãååšããŸããïŒãã®ãšãïŒ
$$\sum_{n = 1}^{40000} na_n$$
ã®ãšãåŸãæå°ã®å€ã解çããŠäžããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/8800 | F | OMC223(F) | 500 | 7 | 15 | [
{
"content": "ã$2$ çŽç· $AB, CD$ ã®äº€ç¹ã $F$ ãšãããšãïŒ$AE = BC$ ã§ããããšãšååšè§ã®å®çãã $\\angle ADE = \\angle FDB$ ã§ããïŒåè§åœ¢ $ABDE$ ãåã«å
æ¥ããããšãã $\\angle AED = \\angle FBD$ ãªã®ã§ïŒ $\\triangle AED \\sim \\triangle FBD$ ã§ããïŒæ¡ä»¶ $BD : DE = 11 : 10$ ãã $BF = 11$ ã§ããïŒæ£ã®å®æ° $x, y$ ãçšããŠ\r\n$$DF = 11xïŒDB = 11yïŒDA = 10xïŒDE = 10y$$\r\nãšè¡šãããšãã§ããïŒããã§è£é¡ãäžããïŒ\r\n\r\n---\r\n\r\n**è£é¡ïŒ**\\\r\nã$2$ ã€ã®äžè§åœ¢ $XYZ, X^{\\prime}Y^{\\prime}Z^{\\prime}$ ãäžããããŠããïŒèŸº $XY, X^{\\prime}Y^{\\prime}$ ã®äžç¹ããããã $M, M^{\\prime}$ ãšãããšãã«\r\n$$\\angle XZM = \\angle X^{\\prime}Z^{\\prime}M^{\\prime}ïŒ\\angle YZM = \\angle Y^{\\prime}Z^{\\prime}M^{\\prime}$$\r\nãæãç«ã€ãªãã°ïŒ$\\triangle XYZ \\sim \\triangle X^{\\prime}Y^{\\prime}Z^{\\prime}$ ã§ããïŒ\r\n\r\n<details><summary>è£é¡ã®èšŒæ<\\/summary>\r\nãäžè§åœ¢ $XYZ, X^{\\prime}Y^{\\prime}Z^{\\prime}$ ã®å€æ¥åããããã $\\Omega, \\Omega^{\\prime}$ ãšãïŒçŽç· $ZM$ ãš $\\Omega$ ã®åšïŒçŽç· $Z^{\\prime}M^{\\prime}$ ãš $\\Omega^{\\prime}$ ã®åšã®äº€ç¹ããããã $W, W^{\\prime}$ ãšããïŒãã ã $W \\neq Z, W^{\\prime} \\neq Z^{\\prime}$ ã§ããïŒïŒ\r\n$$\\angle XZW = \\angle X^{\\prime}Z^{\\prime}W^{\\prime}ïŒ\\angle YZW = \\angle Y^{\\prime}Z^{\\prime}W^{\\prime}$$\r\nãšååšè§ã®å®çãã\r\n$$\r\n\\angle XYW = \\angle X^{\\prime}Y^{\\prime}W^{\\prime}ïŒ\\angle YXW = \\angle Y^{\\prime}X^{\\prime}W^{\\prime}\r\n$$\r\nãæãç«ã€ã®ã§ïŒ$\\triangle XYW \\sim \\triangle X^{\\prime}Y^{\\prime}W^{\\prime}$ ã§ããïŒãã£ãŠ\r\n$$\r\nYW : Y^{\\prime}W^{\\prime} = XY : X^{\\prime}Y^{\\prime} = YM : Y^{\\prime}M^{\\prime}\r\n$$\r\nãš $\\angle MYW = \\angle M^{\\prime}Y^{\\prime}W^{\\prime}$ ãã $\\triangle MYW \\sim \\triangle M^{\\prime}Y^{\\prime}W^{\\prime}$ ãåããã®ã§ïŒ$\\angle MWY = \\angle M^{\\prime}W^{\\prime}Y^{\\prime}$ ã§ããïŒãããšååšè§ã®å®çãã $\\angle ZXY = \\angle Z^{\\prime}X^{\\prime}Y^{\\prime}$ ãåŸãããïŒãŸãïŒä»®å®ããæããã« $\\angle XZY = \\angle X^{\\prime}Z^{\\prime}Y^{\\prime}$ ã§ããïŒä»¥äžã®ããšãã $\\triangle XYZ \\sim \\triangle X^{\\prime}Y^{\\prime}Z^{\\prime}$ ãåŸãïŒ\r\n<\\/details>\r\n\r\n---\r\nãäžè§åœ¢ $ADF, PDE$ ã«ãããŠ\r\n$$AB = BFïŒPQ = QEïŒ\\angle FDB = \\angle QDE$$\r\nãæãç«ã£ãŠããã®ã§ïŒè£é¡ãã $\\triangle ADF \\sim \\triangle PDE$ ãåŸããïŒäžç·ã®é·ãã®æ¯ $DB : DQ$ ã¯ãã®çžäŒŒæ¯ã«çãããã\r\n$$DQ = \\frac{DB \\times DE}{DF} = \\frac{10y^2}{x}$$\r\nã§ããïŒãã£ãŠ $\\dfrac{DQ}{DA} = \\dfrac{y^2}{x^2}$ ã§ããïŒãŸãïŒåè§åœ¢ $ABCD, BCDE$ ã«ããããPtolemyã®å®çãé©çšãããš\r\n$$\r\n\\begin{aligned}\r\nAC &= \\frac{AB \\cdot CD + AD \\cdot BC}{BD} = \\frac{121\\sqrt{10} + 100x}{11y} \\\\\\\\\r\nBE &= \\frac{AB \\cdot DE + AE \\cdot BD}{AD} = \\frac{22y}{x}\r\n\\end{aligned}\r\n$$\r\nãåŸããïŒ$\\angle ADC = \\angle BDE$ ãã $AC = BE$ ããããã®ã§ïŒãã® $2$ åŒãã\r\n$$100x^2 + 121\\sqrt{10}x = 242y^2$$\r\nãåŸãããïŒãã£ãŠïŒ$z = \\dfrac{1}{x}$ ãšãããšã\r\n$$\\dfrac{DQ}{DA} = \\frac{50}{121} + \\frac{\\sqrt{10}}{2}z$$\r\nãšè¡šããïŒæ¹ã¹ãã®å®çãã $DF \\cdot CF = AF \\cdot BF$ ãæãç«ã€ã®ã§ïŒ\r\n$$11x \\cdot 11(x - \\sqrt{10}) = 22 \\cdot 11$$\r\nã§ããïŒãããããšã«æ¬¡ã®æ¹çšåŒãåŸãããïŒ\r\n$$2z^2 + \\sqrt{10}z - 1 = 0$$\r\nãã®æ¹çšåŒã解ãããšã§ $z =\\dfrac{3\\sqrt{2} - \\sqrt{10}}{4}$ ãåŸãããã®ã§ïŒæ±ããå€ã¯\r\n$$\\frac{DQ}{DA} = \\frac{363 \\sqrt{5} - 405}{484}$$\r\nã§ããïŒç¹ã«è§£çãã¹ãå€ã¯ $\\mathbf{1257}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/8800"
},
{
"content": "ãå
¬åŒè§£èª¬ã®è£é¡ãŸã§ã¯åæãšããïŒããªãã¡ $\\triangle FBD \\sim \\triangle AED \\sim \\triangle EQD$ ã§ããïŒ\\\r\nããã£ãŠ $\\dfrac{DB}{DF}=\\dfrac{DE}{DA}=\\dfrac{DQ}{DE}$ ã§ããïŒç¹ã« $\\dfrac{DQ}{DA}=\\left(\\dfrac{DB}{DF}\\right)^2$ ã§ããïŒ\\\r\nãããã§æ¹ããŠå³ãèŠããšïŒæ¹ã¹ãã®å®çïŒ$FB\\cdot FA=FC \\cdot FD$ïŒãã $DF$ ãæ±ããããšãå¯èœã§ããïŒ$DF$ ãæ±ãŸãã°çžäŒŒæ¯ãã $DA$ ãæ±ãŸãïŒ$DA=\\dfrac{10}{11}DF$ïŒïŒããã«äžç·å®çãã $DB$ ãæ±ãŸãã®ã§ïŒããšã¯å°éã«ããããã®é·ããæ±ããŠããã°ããããšã«ãªãïŒ\\\r\nãããã§ïŒ$DF=t$ ãšãã㊠å
ã«æ±ããã¹ãæ¯ãè¡šããŠãããïŒäžç·å®çãã $DB^2=\\dfrac{221}{242}t^2-121$ ãªã®ã§ïŒæ±ããã¹ãæ¯ã¯\r\n$$\\dfrac{DQ}{DA}=\\dfrac{221}{242}-\\dfrac{121}{t^2}$$\r\nãšè¡šãããïŒæåŸã«ïŒæ¹ã¹ãã®å®çãçšã㊠$t=\\dfrac{11\\sqrt{2}}{2}(3+\\sqrt{5})$ ãåŸãïŒãããå
ã»ã©ã®åŒã«ä»£å
¥ããã°ããïŒ",
"text": "çžäŒŒãšäžç·å®çãçšãã",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/8800/550"
},
{
"content": "**è£é¡.** \\\r\nã$2$ ã€ã®äžè§åœ¢ $XYZ,X^{\\prime}Y^{\\prime}Z^{\\prime}$ ãäžããããŠããïŒèŸº $XY,X^{\\prime}Y^{\\prime}$ ã®äžç¹ããããã $M,M^{\\prime}$ ãšãããšãã«ïŒ\r\n$$â XZM = â X^{\\prime}Z^{\\prime}M^{\\prime}, ~ â YZM = â Y^{\\prime}Z^{\\prime}M^{\\prime}$$\r\nãæãç«ã£ãŠãããªãã° $\\triangle XYZ \\sim \\triangle X^{\\prime}Y^{\\prime}Z^{\\prime}$.\r\n---\r\n\r\nãäžè§åœ¢ $ZXM$ ãš $ZYM$ ã«ã€ããŠïŒé¢ç©æ¯ãã\r\n$$\\frac{ZXÃZMÃ\\sinâ XZM}{ZYÃZMÃ\\sinâ YZM} = 1$$\r\nåæ§ã« $Z^{\\prime}X^{\\prime}M^{\\prime}$ ãš $Z^{\\prime}Y^{\\prime}M^{\\prime}$ ã«ã€ããŠïŒ\r\n$$\\frac{Z^{\\prime}X^{\\prime}ÃZ^{\\prime}M^{\\prime}Ã\\sinâ X^{\\prime}Z^{\\prime}M^{\\prime}}{Z^{\\prime}Y^{\\prime}ÃZ^{\\prime}M^{\\prime}Ã\\sinâ Y^{\\prime}Z^{\\prime}M^{\\prime}} = 1$$\r\nãåŸã£ãŠïŒ\r\n$$\\frac{ZXÃZMÃ\\sinâ XZM}{ZYÃZMÃ\\sinâ YZM} = \\frac{Z^{\\prime}X^{\\prime}ÃZ^{\\prime}M^{\\prime}Ã\\sinâ X^{\\prime}Z^{\\prime}M^{\\prime}}{Z^{\\prime}Y^{\\prime}ÃZ^{\\prime}M^{\\prime}Ã\\sinâ Y^{\\prime}Z^{\\prime}M^{\\prime}}$$\r\nã§ïŒä»®å®ãã $\\dfrac{ZX}{ZY} = \\dfrac{Z^{\\prime}X^{\\prime}}{Z^{\\prime}Y^{\\prime}}$ ãåããïŒ$â XZY = â X^{\\prime}Z^{\\prime}Y^{\\prime}$ ã§ããããïŒ\r\n$$\\triangle XYZ \\sim \\triangle X^{\\prime}Y^{\\prime}Z^{\\prime}$$\r\nãåŸãïŒ",
"text": "è£é¡ã®å¥èšŒæ",
"url": "https://onlinemathcontest.com/contests/omc223/editorial/8800/553"
}
] | ãåžäºè§åœ¢ $ABCDE$ ãåã«å
æ¥ããŠããïŒããã«ä»¥äžã®æ¡ä»¶ããã¹ãŠã¿ãããŠããŸãïŒ
$$AB = 11ïŒAE = BC = 10ïŒCD = 11 \sqrt{10}ïŒBD : DE = 11 : 10$$
ããã§ç·å $BD$ äžã«ç¹ $P$ ããšãïŒ$2$ ã€ã®ç·å $PE, DA$ ã®äº€ç¹ã $Q$ ãšãããšããïŒ$PQ = QE$ ãæãç«ã¡ãŸããïŒãã®ãšãïŒ$\dfrac{DQ}{DA}$ ã®å€ãæ±ããŠãã ããïŒ\
ããã ãïŒæ倧å
¬çŽæ°ã $1$ ã§ãã $3$ ã€ã®æ£æŽæ° $a, b, c$ ãšå¹³æ¹å åããããªãæ£æŽæ° $d$ ã«ãã£ãŠ $\dfrac{DQ}{DA} = \dfrac{a \sqrt{d} - b}{c}$ ãšè¡šãããšãã§ããã®ã§ïŒ$a + b + c + d$ ã®å€ã解çããŠãã ããïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/7592 | A | OMCB013(A) | 100 | 337 | 345 | [
{
"content": "ãéè² æŽæ° $n$ ã $4$ ã§å²ã£ãäœããš $5$ ã§å²ã£ãäœãããšãã« $r$ ã§ãããšãããšïŒ$n-r$ 㯠$4$ ã®åæ°ã〠$5$ ã®åæ°ã§ããããïŒ$20$ ã®åæ°ã§ããïŒãŸãïŒ$r$ ãšããŠèããããå€ã¯ $0, 1, 2, 3$ ã§ããïŒ\\\r\nãéã«ïŒ$n=20m+r$ïŒ$m$ ã¯æŽæ°ïŒ$r=0,1,2,3$ïŒãšè¡šããã $n$ 㯠$4$ ã§å²ã£ãäœããš $5$ ã§å²ã£ãäœããçããïŒ\\\r\nãåŸã£ãŠïŒ$20$ ã§å²ã£ãŠ $0,1,2,3$ äœã $100$ æªæºã®éè² æŽæ°ã®æ°ãæ±ããã°ããïŒãã㯠$\\mathbf{20}$ åããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb013/editorial/7592"
}
] | ã次ã®æ¡ä»¶ãæºããæŽæ° $n$ ã¯ããã€ãããŸããïŒ
- $0\leq n\leq 99$
- $n$ ã $4$ ã§å²ã£ãäœããš $n$ ã $5$ ã§å²ã£ãäœãã¯çããïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/9711 | B | OMCB013(B) | 100 | 271 | 324 | [
{
"content": "ã$A$ ããã $x,y,z$ ã®é ã«æããšããïŒãã®ãšãïŒ$B$ ãã㯠$y,z,x$ ãŸã㯠$z,x,y$ ã®é ã«æãå¿
èŠãããïŒåè
ã®å Žå㯠$C$ ãã㯠$x,y,z$ ãŸã㯠$z,x,y$ ãšæãããšã«ãªãïŒåŸè
ã®å Žå㯠$C$ ãã㯠$x,y,z$ ãšæãããšã«ãªãïŒ$A$ ããã®æãæ¹ã¯ $6$ éãããããïŒä»¥äžã«ããæ±ããå Žåã®æ°ã¯ $6 \\times 3= \\mathbf{18} $ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb013/editorial/9711"
}
] | ã$A$ ããïŒ$B$ ããïŒ$C$ ããã® $3$ 人ãã«ã©ãªã±ã«æ¥ãŠããïŒ$A,B,C,A,B,C,A,B,C$ ã®é ã« $1$ æ²ãã€æããŸãïŒæ¬¡ã®æ¡ä»¶ãæºãããããªæ²é ãšããŠããããã®ã¯äœéããããŸããïŒ
- $3$ 人ãšããåã代ããä»°ãã°å°ãããèã®å
ãã® $3$ æ²ãäžåºŠãã€æãïŒ
- $2$ 人以äžãé£ç¶ããŠåãæ²ãæãããšã¯ãªãïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/4694 | C | OMCB013(C) | 200 | 167 | 249 | [
{
"content": "ã$N$ ã®çŽæ°ã®åæ°ã $n$ ãšãïŒ$N$ ã®ä»»æã® $\\sqrt{N}$ 以äžã®çŽæ° $d$ ã«ã€ã㊠$T_d = \\\\{d, N\\/d\\\\}$ ãšããïŒãã®ãšã, çžç°ãªã $2$ æ° $a, b$ ã®ç©ã $N$ ã§ããããšãšïŒãã $\\sqrt{N}$ 以äžã® $N$ ã®çŽæ° $d$ ãååšã㊠$T_d = \\\\{a, b\\\\}$ ãšãªãããšã¯åå€ã§ããïŒ$\\sqrt{N}$ 以äžã® $N$ ã®çŽæ°ã®åæ°ã¯ $\\lceil n\\/2\\rceil$ åã§ããããïŒé³©ã®å·£åçãã $k$ ãšããŠããåŸãæå°å€ã¯ $\\lceil n\\/2\\rceil + 1$ ã§ããïŒããã $102$ ãšçããããšãã $n = 201, 202$ ãåŸãïŒåŸã£ãŠ $N$ ã¯çžç°ãªãçŽ æ° $p, q$ ãçšããŠ\r\n$$N = p^{200}\\quad ãŸãã¯\\quad N = p^{66}q^2 \\quad ãŸãã¯\\quad N = p^{201} \\quad ãŸãã¯\\quad N = p^{100}q$$\r\nãšè¡šãããã®ã§ïŒ$v_2(N)$ ãšããŠããåŸãå€ã¯ $0, 1, 2, 66, 100, 200, 201$ ã§ããïŒåŸã£ãŠè§£çãã¹ãå€ã¯ $\\bf{570}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb013/editorial/4694"
}
] | ãæ£æŽæ° $N$ ã«å¯ŸããŠïŒä»¥äžã®æ¡ä»¶ãæºããæ£æŽæ° $k$ ã®æå°å€ã $102$ ã§ããïŒ
- $k$ 㯠$N$ ã®æ£ã®çŽæ°ã®åæ°ä»¥äžã§ããïŒ
- çžç°ãªã $k$ åã® $N$ ã®æ£ã®çŽæ°ãã©ã®ããã«ãšã£ãŠãïŒãããã®äžã«ç©ã $N$ ã§ãããããªçžç°ãªã $2$ æ°ãååšããïŒ
$v_2(N)$ ã®å€ãšããŠãããããã®ã®ç·åãæ±ããŠãã ããïŒ\
ããã ãïŒæ£æŽæ° $M$ ã«ã€ããŠïŒ$M$ ã $2$ ã§å²ãåããæ倧ã®åæ°ã $v_2(M)$ ãšè¡šããŸãïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/6862 | D | OMCB013(D) | 300 | 172 | 199 | [
{
"content": "ã$\\bmod\\ 5$ ã§èãããšïŒä»»æã®æŽæ° $n$ ã«ã€ããŠïŒ$n^4$ 㯠$0$ ã $1$ ãšçããïŒãŸãïŒ$38671875$ 㯠$5$ ã®åæ°ãªã®ã§ïŒ$x,y,z,w$ ã¯å
šãŠ $5$ ã®åæ°ã§ããïŒäž¡èŸºã $5^4$ ã§å²ããšïŒ\r\n$$(x\\/5)^4 + (y\\/5)^4 + (z\\/5)^4 + (w\\/5)^4 = 61875$$\r\nã§ããïŒå³èŸºã¯åã³ $5$ ã®åæ°ãšãªãã®ã§ïŒåæ§ã®è°è«ãããããšã§ $x\\/5,y\\/5,z\\/5,w\\/5$ ã¯å
šãŠ $5$ ã®åæ°ã§ããïŒåŸã£ãŠïŒ$x = 25a, y=25b, z=25c, w=25d$ ãšããã°\r\n$$a^4+b^4+c^4+d^4 = 99$$\r\nãšãªãïŒãããæºããæ£æŽæ°ã®çµ $(a,b,c,d)$ 㯠$(3,2,1,1)$ ã®äžŠã³æ¿ãã®ã¿ã§ããããšã確èªã§ããã®ã§ïŒæ±ããçã㯠$25(3+2+1+1)\\cdot 12 = \\bf{2100}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb013/editorial/6862"
}
] | ã$$ x^4+y^4+z^4+w^4=38671875$$ãæºããæ£æŽæ°ã®çµ $(x, y, z, w)$ å
šãŠã«ã€ããŠïŒ$x+y+z+w$ ã®ç·åã解çããŠãã ããïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/7608 | E | OMCB013(E) | 300 | 74 | 119 | [
{
"content": "ã$a_2 = 2^8$ ã§ããïŒãŸãïŒä»»æã®æ£ã®æŽæ° $n$ ã«å¯ŸãïŒ\r\n$$a_{n+2} = \\frac{4a_{n+1}^4}{\\prod_{k=1}^{n+1}a_k} = \\frac{4a_{n+1}^4}{4a_{n}^4} = \\frac{a_{n+1}^4}{a_n^4}$$\r\nãæãç«ã€ïŒãã£ãŠïŒ$b_n = \\log_2a_n$ ãšãããšïŒæ°å $\\\\{b_n\\\\}$ 㯠$b_1 = 2, b_2 = 8$ ãã€ä»»æã®æ£ã®æŽæ° $n$ ã«ã€ããŠ\r\n$$b_{n+2} = 4b_{n+1} - 4b_{n}$$\r\nãã¿ããïŒããã解ããšïŒ$b_n = n2^n$ ãåããã®ã§ïŒ$a_n = 2^{n2^{n}}$ ã§ããïŒ\\\r\n$$b_{999} = 999\\cdot2^{999} \\equiv 90\\cdot2^{-1} \\equiv 45 \\pmod{101}$$\r\nããïŒ$\\dfrac{b_{999}-45}{101}$ ãæ£æŽæ°ã§ããããšã«æ°ãã€ãããšïŒ\r\n$$a_{999} = 2^{b_{999}} = 2^{45} \\big((2^{101}-1)+1\\big)^{\\frac{b_{999}-45}{101}} \\equiv 2^{45} \\pmod{2^{101}-1}$$\r\nã§ããïŒãã£ãŠ $r=2^{45}$ ã§ãããã®çŽæ°ã®åæ°ã¯ $\\bf46$ åã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb013/editorial/7608"
}
] | ãæ£ã®æŽæ°ãããªãæ°å $\\{a_n\\}$ 㯠$a_1=4$ ããã³ïŒä»»æã®æ£ã®æŽæ° $n$ ã«å¯ŸããŠä»¥äžãæºãããŸã.
$$\prod_{k=1}^{n+1}a_k=4a_n^4$$
ããã®ãšã $a_{999}$ ã $2^{101} - 1$ ã§å²ã£ãäœãã $r$ ãšããŸãïŒ$r$ ã®æ£ã®çŽæ°ã®åæ°ãæ±ããŠãã ããïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/6450 | F | OMCB013(F) | 400 | 45 | 66 | [
{
"content": "ã**è£é¡**ïŒäžè§åœ¢ $ABC$ ã«ãã㊠$AB:AC=p:q,~ BC=k$ ã§ãããšãïŒãã®é¢ç©ã®æ倧å€ã¯ $\\dfrac{pq}{2|p^2-q^2|}k^2$ ã§ããïŒ\r\n<details><summary> 蚌æ<\\/summary>\r\nã$A$ 㯠$BC$ ã«å¯Ÿãã $p:q$ ã®ã¢ããããŠã¹ã®åäžã«ããã®ã§ïŒ$A$ ãš $BC$ ã®è·é¢ã®æ倧å€ã¯ãã®åã®ååŸïŒããªãã¡ $\\dfrac{pq}{|p^2-q^2|}BC$ ã§ããïŒãããã£ãŠäžè§åœ¢ $ABC$ ã®é¢ç©ã®æ倧å€ã¯ $\\dfrac{pq}{2|p^2-q^2|}k^2$ ã§ããïŒ\r\n<\\/details>\r\nã\\\r\nãè£é¡ã«ããïŒåè§åœ¢ $ABCD$ ã®é¢ç©ã®æ倧å€ã¯\r\n$$\\frac{k^2}{2}\\Big(\\frac{119\\cdot 124}{124^2-119^2}+\\frac{127\\cdot 129}{129^2-127^2} \\Big)=\\frac{k^2}{2}\\Big(\\frac{119\\cdot 124}{3^5\\cdot 5}+\\frac{127\\cdot 129}{2^9} \\Big)$$\r\nã§ããïŒæ¢çŽåæ° $\\dfrac{119\\cdot 124}{3^5\\cdot 5}$ ããã³ $\\dfrac{127\\cdot 129}{2^9}$ ã®åæ¯ã¯äºãã«çŽ ãªã®ã§ïŒãã® $2$ ã€ã®åæ°ã®åãæ¢çŽåæ°ã§è¡šãããšãã®åæ¯ã¯ $2^9\\cdot 3^5\\cdot 5$ ã§ããïŒ\r\nãçµå±ïŒ$2^{10}\\cdot 3^5\\cdot 5$ ã $k^2$ ãå²ãåãã°ããïŒãã®ãã㪠$k$ ã®æå°å€ã¯ $2^5\\cdot 3^3\\cdot 5=\\bf4320$ ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb013/editorial/6450"
}
] | ã$k$ ã¯æ£ã®æŽæ°ãšããŸãïŒèªå·±äº€å·®ãæããªããïŒåžãšã¯éããªãåè§åœ¢ $ABCD$ ã¯æ¬¡ãæºãããŸãïŒ
$$AB:BC=119:124,\quad AD:DC=127:129,\quad AC=k$$
ãã®ãããªåè§åœ¢ $ABCD$ ã®é¢ç©ãšããŠããããæ倧å€ãæŽæ°å€ãšãªããã㪠$k$ ã®æå°å€ãæ±ããŠãã ããïŒ |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/7889 | A | OMC222(A) | 100 | 341 | 346 | [
{
"content": "ã$12$ ã§å²ã£ãäœãã $a$ ãšããã°ïŒæ¡ä»¶ãã¿ããæ£ã®æŽæ°ã¯ $12\\times 2a+a=25a$ ãšè¡šãããšãã§ããïŒãããã£ãŠïŒæ±ããç·åã¯\r\n$$ 25\\times(1+2+\\cdots+11)=\\mathbf{1650}. $$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc222/editorial/7889"
}
] | ãæ£ã®æŽæ°ã§ãã£ãŠïŒ$12$ ã§å²ã£ãåãïŒ$12$ ã§å²ã£ãäœãã®ã¡ããã© $2$ åãšãªããã®ããã¹ãŠæ±ãïŒãããã®ç·åã解çããŠãã ããïŒ |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/4772 | B | OMC222(B) | 200 | 189 | 253 | [
{
"content": "ãäžè§åœ¢ $APQ$ ã¯äºç蟺äžè§åœ¢ãªã®ã§ïŒ$\\angle BAP=\\angle AQB$ ã§ããïŒãã£ãŠïŒäžè§åœ¢ $ABP$ ãš $QBA$ ã¯çžäŒŒã§ããã®ã§ïŒ\r\n$$BP:6=6:(BP+PQ)=6:(BP+5)$$\r\nãæãç«ã€ïŒããã解ãã° $BP=4$ ããããã®ã§ïŒäžè§åœ¢ $ABP$ ã«å¯ŸããäœåŒŠå®çãã $\\cos\\angle{B}=\\dfrac{9}{16},\\tan\\angle{B}=\\dfrac{5\\sqrt7}{9}$ ãåããïŒæ±ããå€ã¯\r\n$$\\Big(6\\times 6\\tan\\angle{B} \\times \\frac{1}{2}\\Big)^2=\\mathbf{700}$$ \r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc222/editorial/4772"
}
] | ã$AB=6,\angle{A}=90^{\circ}$ ã®çŽè§äžè§åœ¢ $ABC$ ã®èŸº $BC$ äžã« $2$ ç¹ $P,Q$ ã $B,P,Q,C$ ã®é ã«äžŠã¶ããã«åããšïŒ$$AP=PQ=5,\quad \angle{BAP}=\angle{PAQ}$$ ãæãç«ã¡ãŸããïŒäžè§åœ¢ $ABC$ ã®é¢ç©ã®äºä¹ãæ±ããŠãã ããïŒ |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/10459 | C | OMC222(C) | 300 | 84 | 115 | [
{
"content": "ã$N = 119$ ãšããïŒæ±ããã¹ãå€ $A$ ã¯ïŒ\r\n$$\r\nA \r\n= \\frac{\\sin 46^\\circ + \\sin 47^\\circ + \\cdots + \\sin 164^\\circ}{\\sin 1^\\circ + \\sin 2^\\circ + \\cdots + \\sin 119^\\circ}\r\n= \\frac{\\displaystyle \\sum_{k=1}^{N} \\sin \\frac{(k+45)\\pi}{180}}{\\displaystyle \\sum_{k=1}^{N} \\sin \\frac{k\\pi}{180}}\r\n$$\r\nãšè¡šããïŒ$\\sin$ ã®å æ³å®çãã $\\sin \\left( x+\\dfrac{\\pi}{4} \\right) = \\dfrac{\\sqrt{2}}{2}(\\sin x + \\cos x)$ ã§ããã®ã§ïŒ\r\n$$\r\nA = \\dfrac{\\sqrt{2}}{2}\\left(1+\\dfrac{\\displaystyle \\sum_{k=1}^{N} \\cos\\frac{k\\pi}{180}}{\\displaystyle \\sum_{k=1}^{N} \\sin \\frac{k\\pi}{180}}\\right)\r\n$$\r\n\r\nã§ããïŒããã§ïŒ\r\n$\\sin x = \\cos \\left( \\dfrac{\\pi}{2}-x \\right)$ ã§ããããšãšïŒåç©ã®åŒããïŒ \r\n$$\r\n\\begin{aligned}\r\n\\dfrac{\\displaystyle \\sum_{k=1}^{N} \\cos \\frac{k\\pi}{180}}{\\displaystyle \\sum_{k=1}^{N} \\sin \\frac{k\\pi}{180}} \r\n&= \\dfrac{\\displaystyle \\sum_{k=1}^{N} \\cos \\frac{k\\pi}{180}}{\\displaystyle \\sum_{k=1}^{N} \\cos \\frac{(90-k)\\pi}{180}} \\\\\\\\\r\n&= \\dfrac{\\dfrac{1}{2}\\displaystyle \\sum_{k=1}^{N} \\bigg(\\cos \\frac{k\\pi}{180}+\\cos \\frac{(N+1-k)\\pi}{180}\\bigg)}{\\dfrac{1}{2}\\displaystyle \\sum_{k=1}^{N} \\bigg(\\cos \\frac{(90-k)\\pi}{180}+\\cos \\frac{(k-N+89)\\pi}{180}\\bigg)}\\\\\\\\\r\n&= \\dfrac{\\displaystyle \\sum_{k=1}^{N} \\cos \\frac{(N+1)\\pi}{360}\\cos \\frac{(2k-N-1)\\pi}{360}}{\\displaystyle \\sum_{k=1}^{N} \\cos \\frac{(179-N)\\pi}{360}\\cos \\frac{(2k-N-1)\\pi}{360}} \\\\\\\\\r\n&= \\frac{\\cos 60^\\circ}{\\cos 30^\\circ} \\\\\\\\\r\n&= \\frac{1}{\\sqrt3}\r\n\\end{aligned}\r\n$$\r\nãšãªãïŒ$A=\\dfrac{\\sqrt3 + 1}{\\sqrt6}$ ãšãªãïŒãã®ãšã\r\n$$A^2 = \\dfrac{\\sqrt3 + 2}{3} = \\dfrac{\\sqrt6}{3}A + \\dfrac13$$\r\nãã\r\n$$ \\frac{2}{3} A^2 = \\left( A^2 - \\frac13 \\right)^2 = A^4 - \\frac23 A^2 + \\frac19 $$\r\nã§ããããïŒ$A$ ã®æå°å€é
åŒ $f$ ã¯\r\n$$\r\nf(x) = x^4-\\frac{4}{3}x^2+\\frac{1}{9}\r\n$$\r\nã§äžããããïŒãããã£ãŠ $|f(10)| = \\dfrac{88801}{9} = 9866.7\\ldots$ ãªã®ã§ïŒè§£çãã¹ãå€ã¯ $\\mathbf{9866}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc222/editorial/10459"
},
{
"content": "ãåé¡æäžã«äžããããŠããå€ã $A$ ãšããïŒ\\\r\nã$\\sin (60\\degree+x\\degree)+\\sin(60\\degree-x\\degree)=2\\sin 60\\degree \\cos x\\degree$ ããã³ $\\sin (105\\degree+x\\degree)+\\sin(105\\degree-x\\degree)=2\\sin 105\\degree \\cos x\\degree$ ãã\r\n$$A=\\dfrac{\\sin 105\\degree(1+2\\cos 1\\degree +\\cdots +2\\cos 59\\degree)}{\\sin 60\\degree(1+2\\cos 1\\degree +\\cdots +2\\cos 59\\degree)}=\\dfrac{\\sin 105\\degree}{\\sin 60\\degree}=\\dfrac{\\sqrt3 +1}{\\sqrt6}$$\r\nãšãªãïŒããšã¯å
¬åŒè§£èª¬ã«åŸã£ãŠæ±ããããïŒ",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc222/editorial/10459/539"
}
] | ã以äžã®å€ã®æå°å€é
åŒã $f$ ãšãããšãïŒ$|f(10)|$ 以äžã®æ倧ã®æŽæ°ã解çããŠãã ããïŒ
$$
\frac{\sin 46^\circ + \sin 47^\circ + \sin 48^\circ + \cdots + \sin 164^\circ}{\sin 1^\circ + \sin 2^\circ + \sin 3^\circ + \cdots + \sin 119^\circ}
$$
<details><summary>æå°å€é
åŒã«ã€ããŠ<\/summary>
ãè€çŽ æ° $\alpha$ ã«ã€ããŠïŒ$\alpha$ ãæ ¹ã«ãã€æçæ°ä¿æ°å€é
åŒãååšãããšãïŒãã®ãã¡æ¬¡æ°ãæå°ã§ããïŒãã€æé«æ¬¡ã®ä¿æ°ã $1$ ã§ãããã®ãïŒ $\alpha$ ã®**æå°å€é
åŒ**ãšããã§ã¯ãã³ãŸãïŒ
<\/details> |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/9719 | D | OMC222(D) | 300 | 177 | 248 | [
{
"content": "ã$1!$ ãã $3! = 6$ åæã $2$ æãŸã§çšããããšã§ïŒ$1$ åãã $18$ åãŸã§ $1$ åå»ã¿ã§ã¡ããã©æãããšãã§ããïŒäžæ¹ã§ïŒ$n \\geq 4$ ã«ãããŠã¯ïŒåž°çŽçã«\r\n$$ 2\\cdot1!+2\\cdot2!+\\cdots+2\\cdot(n-1)! \\lt n!$$\r\nã瀺ãããšãã§ããããïŒ$n!$ åæªæºã®ãæã $2$ æãã€çšããéé¡ããã $n!$ ã®æ¹ã倧ãããªãïŒãããã£ãŠïŒäœ¿çšãã $n!$ åæã®ææ°ããšã«ç°ãªãåèšéé¡ã察å¿ããïŒãããã£ãŠ $n \\geq 4$ ã®ãšãïŒ$n!$ åæãŸã§ã䜿çšããŠäœãããšã®ã§ããéé¡ã®çš®é¡ã¯ïŒ$19 \\cdot 3^{n-3}-1$ çš®é¡ã§ããïŒããã㯠$0 \\le C \\le 18$ ããã³ $c_4, c_5, \\ldots, c_n \\in \\\\{0, 1, 2 \\\\}$ïŒãã¹ãŠã $0$ ãšã¯ãªããªãïŒã«ãã£ãŠ\r\n$$ C + c_4 \\cdot 4! + c_5 \\cdot 5! + \\cdots + c_n \\cdot n! $$\r\nãšäžæã«è¡šç€ºãããïŒããã«ïŒããã¯ã¡ããã©æ¯æããéé¡ã®ãã¡ïŒå°ããæ¹ãã\r\n$$ C + c_4 \\cdot 19 \\cdot 3^0 + c_5 \\cdot 19 \\cdot 3^1 + \\cdots + c_n \\cdot 19 \\cdot 3^{n-4} $$\r\nçªç®ã®ãã®ã§ããããšããããïŒããã $1000$ ã«çãããªãã®ã¯ïŒ$n=7$ ãã€\r\n$$ (C, c_4, c_5, c_6, c_7) = (12, 1, 2, 2, 1) $$\r\nã®ãšãã§ããïŒãã®ãšãïŒæ¯æãããéé¡ã¯\r\n$$ 12 + 1 \\cdot 4! + 2 \\cdot 5! + 2 \\cdot 6! + 1 \\cdot 7! = \\mathbf{6756}$$\r\nåã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc222/editorial/9719"
}
] | ãOMCåœã§ã¯ïŒ$n=1,2,\ldots,1000$ ããããã«å¯ŸãïŒ$n!$ åæãé貚ãšããŠæµéããŠããŸãïŒOMCåœã«ãã£ãŠããããªãã¯ïŒããããã®ãæã $2$ æãã€æã£ãŠããŸãïŒããªããã¡ããã©æãããšã®ã§ããéé¡ãšã㊠$1000$ çªç®ã«å°ããå€ãçããŠãã ããïŒ\
ããã ãïŒã¡ããã©æãããšã®ã§ããéé¡ãšããŠïŒ$0$ åã¯å«ããŸããïŒ |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/5667 | E | OMC222(E) | 400 | 14 | 26 | [
{
"content": "ãäžèŸº $2$ ã®æ£ $36$ è§åœ¢ $C_0C_1C_2\\cdots C_{35}$ ãèãïŒãã®äžå¿ã $O$ ãšããïŒããã§äžè§åœ¢ $C_0 C_1 O$ ã®é¢ç©ã $U$ ãšããã°ïŒäžèŸº $2$ ã®æ£ $36$ è§åœ¢ã®é¢ç©ã¯ $36U$ ã§ããé·æ¹åœ¢ $C_{0}C_{1}C_{18}C_{19}$ ã®é¢ç©ã¯ $4U$ ã§ããïŒãã®ãšãïŒå€è§åœ¢$A$ , $B$ ãåå²ãäžèŸº $2$ ã®æ£ $36$ è§åœ¢ã«ã¯ãããšå€è§åœ¢ $A$ ãå·Šå³ïŒ$B$ ãå³å³ã®ç¶²æãã®ããã«ãªãã®ã§ïŒ\r\n$$S=36U - 4U\\cdot 2 + 2\\cdot 2 = 28U + 4$$\r\n$$T=36U - 4U\\cdot 3 + 2\\sqrt{3} \\cdot 3 \\cdot \\frac{1}{2} \\cdot 2 = 24U + 6\\sqrt{3}$$\r\nããã£ãŠïŒ$\\lvert 6S-7T\\rvert = 42\\sqrt{3} - 24$ ã§ããã®ã§ïŒç¹ã«è§£çãã¹ãå€ã¯ $\\mathbf{69}$ ã§ããïŒ\r\n",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc222/editorial/5667"
}
] | ãåžå€è§åœ¢ $A = A_{0}A_{1}\cdots A_{31}$ ãš $B = B_{0}B_{1}\cdots B_{29}$ ã¯ãããã以äžã®æ¡ä»¶ãã¿ãããŸãïŒ
- å€è§åœ¢ $A, B$ ã®èŸºã®é·ãã¯ãã¹ãŠ $2$ ã§ããïŒ
- $i$ ã $8$ ã®åæ°ã§ãããšã $\angle {A_i} = 160^\circ$ïŒ$i$ ã $8$ ã®åæ°ã§ãªããšã $\angle {A_i} = 170^\circ$ïŒ
- $j$ ã $5$ ã®åæ°ã§ãããšã $\angle {B_j} = 160^\circ$ïŒ$j$ ã $5$ ã®åæ°ã§ãªããšã $\angle {B_j} = 170^\circ$ïŒ
ãå€è§åœ¢ $A,B$ ã®é¢ç©ããããã $S,T$ ãšãããšãïŒ$\lvert 6S-7T \rvert = a\sqrt{b} - c$ ($a,b,c$ ã¯æ£ã®æŽæ°ïŒ$b$ ã¯å¹³æ¹å åãæããªã) ãšè¡šããã®ã§ïŒ$a+b+c$ ãæ±ããŠãã ããïŒ |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/9299 | F | OMC222(F) | 500 | 51 | 132 | [
{
"content": "ãåé¡æäžã«ãã $2255222255255552$ ã $A$ ãšããïŒ$A$ ã®äž $m$ æ¡ $(1\\leq m\\leq 15)$ ãããªãæŽæ°ã¯ $2^m$ ã®åæ°ãªã®ã§ïŒ$m+1$ æ¡ä»¥äžã®æŽæ°ã§äž $m$ æ¡ã $A$ ãšäžèŽã $m+1$ æ¡ç®ãç°ãªããã®ã¯ $2$ ã§ã¡ããã© $m$ åå²ãããšãã§ããïŒ$m=0$ ã§ãæå³ãæã€ãïŒããã¯ããªãã¡å¥æ°ã§ãããšããæå³ãªã®ã§ïŒèæ
®ããªããŠããïŒïŒ\r\n$m+1$ æ¡ä»¥äž $16$ æ¡ä»¥äžã®æŽæ°ã§äž $m$ æ¡ã $A$ ãšäžèŽã $m+1$ æ¡ç®ãç°ãªãæŽæ°ã¯ $2^{16-m}-1$ åããã®ã§ïŒãããã®ç©ã $2$ ã§å²ããåæ°ã¯ $\\sum_{m=1}^{15} m(2^{16-m}-1)$ ã§è¡šãããïŒ\\\r\nã$A$ ã®äž $m$ æ¡ãããªãæŽæ°ã $2$ ã§å²ããåæ°ã¯ïŒ$A$ ã® $1$ æ¡ç®ãã $k$ æ¡ç®ãš $m+1$ æ¡ç®ãã $m+k$ æ¡ç®ãäžèŽãããã㪠$k$ ã®æå€§å€ $f(m)$ ãçšã㊠$m+f(m)$ åãšãªãïŒ$1$ æ¡ç®ãš $m+1$ æ¡ç®ãç°ãªãå Žå㯠$f(m)=0$ ãšããïŒïŒå®éã«èšç®ããã°ïŒ$f(5)=f(11)=3, f(8)=f(9)=f(10)=f(14)=f(15)=1$ ã§ïŒä»ã¯å
šãŠ $0$ ã§ããïŒ\\\r\nãäžèšãš $A$ 㧠$2$ ãš $5$ ãããªã $16$ æ¡ä»¥äžã®æ£ã®æŽæ°ã¯ç¶²çŸ
ãããã®ã§ïŒæ±ããå€ã¯\r\n$$\\sum_{m=1}^{15} \\big( m(2^{16-m}-1)+m+f(m)\\big)+17=\\sum_{m=1}^{15}(mÃ2^{16-m}) +28=(2^{17}-34)+28=\\mathbf{131066}.$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc222/editorial/9299"
}
] | ãåæ¡ã $2$ ãŸã㯠$5$ ã§ãã $16$ æ¡ä»¥äžã®æ£æŽæ°ã®ãã¡ïŒ$2$ ã§å²ãåããåæ°ãæãå€ãã®ã¯
$$2255222255255552$$
ã®ãã äžã€ã§ïŒãã㯠$2$ ã§æ倧 $17$ åå²ãåãããšãã§ããŸãïŒ\
ãåæ¡ã®æ°ã $2$ ãŸã㯠$5$ ãããªã $16$ æ¡ä»¥äžã®æ£æŽæ°ãã¹ãŠã®ç©ã $N$ ãšãããšãïŒ$N$ ã $2$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠãã ããïŒ |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8033 | A | OMCE004(A) | 300 | 167 | 198 | [
{
"content": "ãäžæ¹ãæã£ãŠããå¥æ°ã®ã«ãŒããšããäžæ¹ãæã£ãŠããå¶æ°ã®ã«ãŒãã®ææ°ã¯çããïŒãŸãïŒ$1,2,\\ldots,8$ ã®ãã¡ $2$ æ°ãäºãã«çŽ ã§ãªãã®ã¯ïŒãšãã«å¶æ°ã§ããå ŽåãïŒ$3$ ãš $6$ ã®ã¿ã§ããïŒãŸãããã«ããïŒäžæ¹ã $3$ ãš $6$ ãåæã«æã£ãŠãããšãã«ã¯ïŒ$B$ ãã㯠$A$ ãããçŽåã«åºããæ°ãšå¶å¥ã®ç°ãªããã®ãä»»æã«åºãããšã§å¿
ãåãŠããšãããïŒ\\\r\nã$A$ ããã $3$ ãïŒ$B$ ããã $6$ ãæã£ãŠãããšããïŒã㟠$A$ ããã $3$ ã®ä»ã«å¥æ°ãæã£ãŠãããšãïŒããªãã¡ $B$ ããã $6$ ã®ä»ã«å¶æ°ãæã£ãŠãããšãïŒïŒ$B$ ããã¯ïŒ$3$ ã®åŸã« $6$ ãåºããªãããã«ããã°ïŒäžãšåãæŠç¥ã§åã€ããšãã§ããïŒããã§ãªããšãïŒããªãã¡ $A$ ããã $2348$ ãïŒ$B$ ããã $1567$ ãæã£ãŠãããšãã¯ïŒ$A$ ãã㯠$3$ ãæåŸãŸã§æ®ãã° $B$ ãã㯠$6$ ãæ®ãããšã«ãªãè² ããïŒ\\\r\nãéã« $A$ ããã $6$ ãïŒ$B$ ããã $3$ ãæã£ãŠãããšãããšïŒåæ§ã«ã㊠$A$ ããã $1567$ ãïŒ$B$ ããã $2348$ ãæã£ãŠãããšãã®ã¿æ€èšããã°ããïŒãã®ãšãïŒ$A$ ããã¯ãŸã $6$ ãåºãã°ããïŒ\\\r\nã以äžã«ããïŒæ±ããã¹ãå€ã¯ $8432 + 7651 = \\mathbf {16083}$ ãšåããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce004/editorial/8033"
}
] | ã$1$ ãã $8$ ãŸã§ã®æŽæ°ã®ãã¡äžã€ãæžãããã«ãŒãããããã $1$ æãã€ãããŸãïŒãããã $4$ æãã€ã«åã㊠$A$ ãããš $B$ ããã«é
ãïŒä»¥äžã®ãããªã²ãŒã ãè¡ããŸãïŒ
- $A$ ãããå
æïŒ$B$ ãããåŸæãšããŠïŒé
ãããã«ãŒããã亀äºã« $1$ æãã€åºããŠããïŒäžåºŠåºããã«ãŒãã¯åã³åºããªãïŒ
- çžæãçŽåã«åºããã«ãŒãã«æžãããæŽæ°ãšäºãã«çŽ ãªæŽæ°ãæžãããã«ãŒããåºãããšãã§ããïŒãã®ã«ãŒã«ã«åŸã£ãŠåºããã«ãŒããç¡ããªã£ãããã®æç¹ã§ã²ãŒã ãçµäºããïŒ
- æåŸã«ã«ãŒããåºãã人ã®åã¡ãšããïŒç¹ã«ïŒãã¹ãŠã®ã«ãŒããåºãåã£ãã $B$ ããã®åã¡ã§ããïŒïŒ
ãããšïŒäž¡è
ãæåãå°œãããå Žåã«ã¯ $A$ ããã®å¿
åãšãªããŸããïŒãã®ãšãïŒ$A$ ããã«é
ããã $4$ æã®ã«ãŒããéé ã«äžŠã¹ãŠã§ãã $4$ æ¡ã®æŽæ°ãšããŠãããããã®ã®ç·åãæ±ããŠãã ããïŒ\
ããã ãïŒ$2$ 人ã®æã£ãŠããã«ãŒãã¯ã€ãã«çžæã«é瀺ãããŠãããã®ãšããŸãïŒ
<details> <summary>
解ç圢åŒã«ã€ããŠ
<\/summary>
ãããšãã° $A$ ããã« $1, 2, 3, 4$ ã®ã«ãŒããé
ããããšãïŒ$A$ ããã«é
ããã $4$ æã®ã«ãŒããéé ã«äžŠã¹ãŠã§ãã $4$ æ¡ã®æŽæ° $N$ 㯠$N = 4321$ ã§ãïŒ$A$ ãããå¿
åãšãªããã¹ãŠã®ã«ãŒãã®é
ãæ¹ã«ã€ããŠïŒãã®ãã㪠$N$ ã®ç·åãæ±ããŠãã ããïŒ
<\/details> |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8291 | B | OMCE004(B) | 500 | 49 | 89 | [
{
"content": "ããŸãïŒåããŒã«ãç¬ç«ã«ãããã $\\dfrac{1}{3}$ ã®ç¢ºçã§èµ€ïŒéïŒçœã«å¡ã£ãŠãããšèããããšã§ïŒæ±ããã¹ã㯠$RBW$ ã®æåŸ
å€ãšèšãæããããšãã§ããïŒãŸãïŒå¡ãæ¹ãåºå®ãããšãã® $RBW$ ã®å€ã¯ïŒé£ãåãããŒã«å士ã®é åºä»ãã $3$ ã€ã®çµ $(A_1,A_2,A_3)$ ã§ãã£ãŠïŒ$A_1, A_2, A_3$ ã«å«ãŸãã $2$ ã€ã®ããŒã«ãããããèµ€ãšèµ€ïŒéãšéïŒçœãšçœã§å¡ãããŠãããããªãã®ã®æ°ã«çããïŒ\r\n$ \\\\\\ $ãããã§, $(A_1, A_2, A_3)$ ãéžãã æ, $A_1, A_2, A_3$ ã«å«ãŸãã $2$ ã€ã®ããŒã«ãããããèµ€ãšèµ€, éãšé, çœãšçœã§å¡ãããŠãããšã $(A_1, A_2, A_3)$ ã®ã¹ã³ã¢ã $1$ , ããã§ãªããšã $(A_1, A_2, A_3)$ ã®ã¹ã³ã¢ã $0$ ãšå®ããããšã§, å
šãŠã® $(A_1,A_2,A_3)$ ã«å¯Ÿããã¹ã³ã¢ã®åã $RBW$ ã«çãããªã, æ±ããã¹ãã¯å
šãŠã® $(A_1,A_2,A_3)$ ã«å¯Ÿããã¹ã³ã¢ã®åã®æåŸ
å€ãšåãã. ããã¯, $(A_1,A_2,A_3)$ ã®ã¹ã³ã¢ã®æåŸ
å€ã®åã«çãããã, åçµã®ã¹ã³ã¢ã®æåŸ
å€ãèããã°ãããšåãã.\r\n$ \\\\\\ $ããŸã, $A_1, A_2, A_3$ ãå
±ééšåãæã€ãšãã¯å¿
ãã¹ã³ã¢ã $0$ ãšãªããã, ã¹ã³ã¢ã®æåŸ
å€ã¯ $0$ ã§ãã. äžæ¹, $A_1, A_2, A_3$ ãå
±ééšåãæããªããšãã¯, ã¹ã³ã¢ã$1$ ã«ãªã確çã $\\left( \\dfrac{1}{3} \\right)^6 = \\dfrac{1}{729}$ ã§ããããšãã, ã¹ã³ã¢ã®æåŸ
å€ã¯ $\\dfrac{1}{729}$ ã§ãã. ãã£ãŠ, $A_1, A_2, A_3$ ãå
±ééšåãæããªããã㪠$(A_1,A_2,A_3)$ ã®æ°ãæ±ããã°ãããšåãã. \r\n$ \\\\\\$ ãããã§, ãŸã $A_1$ ãåºå®ãããšãã® $A_2, A_3$ ã®éžã³æ¹ãèãã. ããã¯, $2022$ åã®ããŒã«ã暪äžåã«äžŠãã§ãããšãã«, åºå¥ã§ããé£ãåãããŒã«å士ã®çµã $2$ ã€éãªããªãããã«éžã¶æ¹æ³ã«çãã, ãã㯠$ {}\\_{2020}\\mathrm{C}\\_{2} à 2 = 4078380$ éããã. ãŸã, $A_1$ ã®éžã³æ¹ã¯ $2024$ éããããã, æ¡ä»¶ãæºãã $(A_1,A_2,A_3)$ ã®æ°ã¯å
šéšã§ $4078380 à 2024 = 8254641120$ éããããšåãã. ãã£ãŠ, å
šãŠã®$(A_1,A_2,A_3)$ ã«å¯Ÿããã¹ã³ã¢ã®æåŸ
å€ã®å㯠$8254641120 à \\dfrac{1}{729} = \\dfrac{2751547040}{243}$ ãšãªããã, ç¹ã«è§£çãã¹ãå€ã¯ $\\mathbf{2751547283}$ ãšãªã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce004/editorial/8291"
},
{
"content": "ãæçµçã«ããããšã¯å€ãããŸãããïŒèãæ¹ã ãå¥è§£ã§ãïŒããŸã«ããïŒåœ¢å®¹ççŸïŒèŠããã¯ããã¯ã§ãïŒ\r\n\r\n---\r\n\r\nãéèŠãªã®ã¯ $RBW$ ã®**çµã¿åããçãªæå³**ãèããããšã§ãïŒ\\\r\nãããŒã«ã«é©åœã« $1$ ãã $2024$ ãŸã§ã®çªå·ãä»ããŸãïŒãŸã䟿å®äžïŒããŒã« $2025$ ã¯ããŒã« $1$ ãè¡šããã®ãšããŸãïŒïŒãã®ãšãïŒããå¡ãæ¹ã«ããã $RBW$ ã¯æ¬¡ã®å Žåã®æ°ã«çããããšãåãããŸãïŒ\r\n- $1$ ä»¥äž $2024$ 以äžã®æŽæ°ã®çµ $(r,b,w)$ ã§ãã£ãŠïŒä»¥äžãæºãããã®ã®åæ°ïŒ\r\n - ããŒã« $r$ ãš $r+1$ ã¯ãšãã«èµ€è²ã§ããïŒ\r\n - ããŒã« $b$ ãš $b+1$ ã¯ãšãã«éè²ã§ããïŒ\r\n - ããŒã« $w$ ãš $w+1$ ã¯ãšãã«çœè²ã§ããïŒ\r\n\r\nããã¯ãããããç©ã®æ³åãããç°¡åã«åãããŸãïŒçœäžžã®ç®æ¡æžãããããã $R,B,W$ éãããããã§ãïŒ\\\r\nããã®å Žåã®æ°ããã¹ãŠã®å¡ãæ¹ã«ã€ããŠè¶³ãåãããã°ïŒããã $3^{2024}$ ã§å²ãããšã§å¹³åãåºããŸãïŒä»¥äžã¯ãããèããŸãããïŒ\r\n\r\n---\r\n\r\nãããã§**䞻客転å**ãšåŒã°ãããã¯ããã¯ã䜿ããŸãïŒã€ãŸãïŒã足ãåãããéçšã«ãããŠããçµ $(r,b,w)$ ã¯äœåå¯äžãããïŒããèããŠããã足ãåãããŠãå€ãããŸããïŒæ¬¡ã®ããšãããã«åãããŸãïŒ\r\n- ããŒã« $r,r+1, b, b+1, w, w+1$ ããã¹ãŠç°ãªããšãïŒããããã®ããŒã«ã察å¿ããè²ã§å¡ãïŒãã以å€ã¯èªç±ã«å¡ãå¡ãæ¹ããã¹ãŠã $(r,b,w)$ ãå«ãããïŒãã®çµã¯ $3^{2018}$ åå¯äžããïŒ\r\n- ããã§ãªããšãïŒå¯äžãããããªå¡ãæ¹ã¯ãªãïŒè¢«ã£ãŠããããŒã«ã¯åæã« $2$ è²ã§å¡ããªããã°ãªããªããªãïŒïŒãã£ãŠå¯äžã¯ $0$ åïŒ\r\n\r\nåŸã£ãŠïŒèããã¹ãããšã¯ãããŒã« $r,r+1,b,b+1,w,w+1$ ããã¹ãŠç°ãªããã㪠$r,b,w$ ã¯ããã€ãããïŒãã«çµãããŸããïŒãªããªãïŒãããåããã°ïŒå¯äžã®åèšã¯ãã® $3^{2018}$ åã§ããïŒãããæ±ããŠããç·åã§ããããã§ãïŒ\r\n\r\n---\r\n\r\nãããšã¯ $r$ ãåºå®ãããªã©ããŠïŒä»èããã¹ãããšã®çã㯠$2024 \\times (1+2+\\cdots+2019)\\times 2$ ã ãšåããã®ã§ïŒããã§è§£ããŸããïŒ",
"text": "å¥è§£ïŒå
žåçïŒïŒ",
"url": "https://onlinemathcontest.com/contests/omce004/editorial/8291/537"
}
] | ãååšäžã« $2024$ åã®äºãã«åºå¥ã§ããããŒã«ãããé åºã§äžŠãã§ããŸãïŒãããã®ããŒã«ãããããèµ€è²ïŒéè²ïŒçœè²ã®ãããã $1$ è²ã§å¡ã£ãŠãããŸãïŒãã®ãšãïŒé£ãåã $2$ ã€ã®ããŒã«ã®çµã§ãã£ãŠïŒäž¡æ¹ãšãèµ€è²ã§å¡ããããã®ã®æ°ã $R$ïŒäž¡æ¹ãšãéè²ã§å¡ããããã®ã®æ°ã $B$ïŒäž¡æ¹ãšãçœè²ã§å¡ããããã®ã®æ°ã $W$ ãšããŸãïŒããŒã«ãå¡ãæ¹æ³ã¯å
šéšã§ $3^{2024}$ éããããŸããïŒãããå
šãŠã«å¯Ÿãã $RBW$ ã®çžå å¹³åãæ±ããŠäžããïŒãã ãïŒçãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\cfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠäžãã. |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/10686 | C | OMCE004(C) | 500 | 26 | 57 | [
{
"content": "ã$\\angle ACB = \\theta$ ãšããïŒãããšïŒ$4$ ç¹ $A, X, D, C$ ã®å
±åãã $\\angle AXD = 180^\\circ - \\theta$ ãåŸãïŒ$\\angle AXC = \\angle ADC = 90^\\circ$ ãã $\\angle DXC = 90^\\circ - \\theta$ ãåŸãããïŒãŸãïŒ$4$ ç¹ $A, O, X, B$ ã®å
±åãã $\\angle BXE = \\angle BAO = 90^\\circ - \\theta$ ãåããã®ã§ïŒ\r\n$$\\angle BXC = 360^\\circ - \\angle BXA - \\angle AXC = 360^\\circ - \\angle BOA - \\angle ADC = 270^\\circ - 2\\theta$$\r\nããïŒ$\\angle EXD = 90^\\circ$ ãåŸãïŒããããïŒ\r\n$$\\angle EXD = \\angle CXA = 90^\\circ, \\quad \\angle XDE = 180^\\circ - \\angle XDC = \\angle XAC$$\r\nããïŒäžè§åœ¢ $EXD$ ãšäžè§åœ¢ $CXA$ ãå転çžäŒŒã®é¢ä¿ã«ããããšãåããïŒãã£ãŠïŒäžè§åœ¢ $EXC$ ãšäžè§åœ¢ $DXA$ ãå転çžäŒŒã®é¢ä¿ã«ããïŒ$\\angle BXE = \\angle DXC, \\ \\angle BXD = \\angle EXC$ ããïŒé¢ç©æ¯ãèããããšã§ïŒ\r\n$$BX \\cdot EX : CX \\cdot DX = 2 : 1, \\quad BX \\cdot DX : CX \\cdot EX = 7 : 5$$\r\nãåŸãããïŒãã£ãŠ $EX : DX = \\sqrt{10} : \\sqrt{7}$ ã§ããïŒäžå¹³æ¹ã®å®çãã $DX = \\dfrac{3\\sqrt7}{\\sqrt{17}}$ ã§ããïŒãŸãäžè§åœ¢ $EXC$ ãšäžè§åœ¢ $DXA$ ã®çžäŒŒæ¯ã $\\sqrt{10} : \\sqrt{7}$ ãªã®ã§ïŒ$AD = \\dfrac{\\sqrt{70}}{2}$ ãåããïŒäžå¹³æ¹ã®å®çããïŒ$AC = \\dfrac{ \\sqrt{86}}{2}$ ãšãªãïŒãããŠïŒ\r\n$$AX : AC = DX : DE = \\sqrt{7} : \\sqrt{17}$$\r\nããïŒ$AX^2 = \\dfrac{301}{34}$ ãåããïŒç¹ã«ïŒè§£çãã¹ãå€ã¯ $\\mathbf{335}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce004/editorial/10686"
}
] | ãéè§äžè§åœ¢ $ABC$ ã®å€å¿ã $O$ïŒ$A$ ãã蟺 $BC$ ã«äžãããåç·ã®è¶³ã $D$ ãšããŸãïŒäžè§åœ¢ $AOB$ ã®å€æ¥åãšäžè§åœ¢ $ADC$ ã®å€æ¥åã¯äžè§åœ¢ $ABC$ ã®å
éšã®ç¹ $X (\neq A)$ ã§äº€ãããŸããïŒããã«çŽç· $OX$ ãšçŽç· $BC$ ã®äº€ç¹ã $E$ ãšãããšïŒ$4$ ç¹ $B, E, D, C$ ã¯ãã®é ã«äžŠã³ïŒ
$$BE = 4, \quad ED = 3, \quad DC = 2$$
ãæç«ããŸããïŒãã®ãšã $AX^{2}$ ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠãã ããïŒ |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8808 | D | OMCE004(D) | 600 | 30 | 79 | [
{
"content": "ã$$\\sum_{k = 0}^{1012} 2024^ {1012-k} x^{2k} = \\dfrac{x^{2026} - 2024^{1013}}{x^{2} - 2024}$$ \r\nã§ããããïŒ$Ï = \\cos \\dfrac{2\\pi}{2026} + i \\sin \\dfrac{2\\pi}{2026}$ ãšãããšïŒæ¹çšåŒãæ〠$2024$ åã®è€çŽ æ°è§£ã¯ \r\n$$ \\pm \\sqrt{2024} à Ï^{k} \\quad (1 \\leq k \\leq 1012) $$\r\n\r\nãšè¡šãããããšãåããïŒããã§ïŒ$Ï^{k}$ ãå®æ°ã«ãªãããã®å¿
èŠååæ¡ä»¶ã¯ïŒ$\\dfrac{2k}{2026} = \\dfrac{k}{1013}$ ãæŽæ°ã«ãªãããšïŒããªãã¡ $k$ ã $1013$ ã®åæ°ã«ãªãããšã§ããïŒãŸã, $|Ï^{k}| = |Ï|^{k} = 1$ ã§ããããšããïŒãã㯠$\\omega^k$ ãæŽæ°ã«ãªãããã®å¿
èŠååæ¡ä»¶ã§ããããšãåããïŒãŸã, $\\sqrt{2024}$ ãç¡çæ°ã§ããããšããïŒãã®åé¡ã¯æ¬¡ã®ããã«èšãæããããïŒ\r\n\r\n- å
šéšã§ $2024$ æã®ã«ãŒããããïŒããããã«ã¯ $1$ ã€ãã€æŽæ°ãæžãããŠããïŒãŸã, $1$ ãã $1012$ ãŸã§ã®å
šãŠã®æŽæ°ã«ã€ããŠïŒãããæžãããã«ãŒã㯠$2$ æãã€ååšããïŒå
šãŠã®ã«ãŒããåºå¥ãããšãïŒãããã®ã«ãŒãã®äžãã $1$ æ以äžãã€å¶æ°æéžãã§æžãããæŽæ°ã®åã $1013$ ã®åæ°ã«ãªãããã«ããæ¹æ³ã¯äœéããããïŒ\r\n\r\nããŸãïŒ$1$ æ以äžãšããæ¡ä»¶ããªãããŠèããïŒã«ãŒããå¶æ°æéžãã§åã $1013$ ã®åæ°ã«ãªãããã«ããå Žåã®æ°ã $A$ïŒå¥æ°æéžãã§åã $1013$ ã®åæ°ã«ãªãããã«ããå Žåã®æ°ã $B$ ãšããïŒãã®ãšãïŒ$A+B$ 㯠$$\\sum_{n = 0}^{\\infty} \\ [x^{1013n}] \\ (x + 1)^{2}(x^{2} + 1)^{2}\\cdots(x^{1012} + 1)^{2}$$\r\nãšè¡šãããããšãåããïŒãŸãïŒ$A-B$ 㯠\r\n$$\\sum_{n = 0}^{\\infty} \\ [x^{1013n}] \\ (x - 1)^{2}(x^{2} - 1)^{2}\\cdots(x^{1012} - 1)^{2}$$\r\nãšè¡šãããããšãåãã ( $-1$ ãéžã°ããåæ°ãèããããšã«ããïŒ$B$ ã«å¯Ÿå¿ããé
ã®ã¿ããã¹ãŠ $-1$ åãããããšã確èªãã)ïŒãã ãå€é
åŒ $p$ ãšéè² æŽæ° $d$ ã«ã€ã㊠$[x^d]p(x)$ 㧠$p(x)$ ã® $x^d$ ã®ä¿æ°ãè¡šãïŒä»¥äž\r\n$$ f(x) = (x + 1)^{2}(x^{2} + 1)^{2}\\cdots(x^{1012} + 1)^{2} $$\r\n$$ g(x) = (x - 1)^{2}(x^{2} - 1)^{2}\\cdots(x^{1012} - 1)^{2} $$\r\nãšããïŒããã§ïŒ$Ï^{\\prime}$ ã $Ï^{\\prime} = \\cos \\dfrac{2\\pi}{1013} + i \\ \\sin \\dfrac{2\\pi}{1013}$ ã§å®ãããšãïŒæŽæ° $k$ ã«å¯Ÿã㊠\r\n$$1+{Ï^{\\prime}}^{k}+{Ï^{\\prime}}^{2k}+\\cdots+{Ï^{\\prime}}^{1012k} = \\begin{cases} 0 & (1013 \\nmid k) \\\\\\\\ 1013 & (1013 \\mid k) \\end{cases}$$\r\nã§ããããšããïŒ\r\n$$A+B = \\dfrac{1}{1013} \\left( f(1) + f(Ï^{\\prime}) + f({Ï^{\\prime}}^{2}) + \\cdots + f({Ï^{\\prime}}^{1012}) \\right)$$\r\n$$A-B = \\dfrac{1}{1013} \\left( g(1) + g(Ï^{\\prime}) + g({Ï^{\\prime}}^{2}) + \\cdots + g({Ï^{\\prime}}^{1012}) \\right)$$\r\n\r\nãæãç«ã€ããšãåããïŒããŸïŒ\r\n$$ F(x) = (x - Ï^{\\prime})(x - {Ï^{\\prime}}^{2})\\cdots(x - {Ï^{\\prime}}^{1012}) = x^{1012} + x^{1011} + \\cdots + x + 1 $$\r\nãšå®çŸ©ãããšïŒæŽæ° $1 \\le k \\le 1012$ ã«å¯Ÿã㊠$ f({Ï^{\\prime}}^{k}) = F(-1)^2 = 1$ ããã³ $ g({Ï^{\\prime}}^{k}) = F(1)^2 = 1013^2 $ããããïŒãããš $f(1) = 2^{2024}, g(1) = 0$ ãåãããŠïŒ\r\n$$ A+B = \\dfrac{2^{2024} + 1012}{1013}, \\quad A-B = 1012 \\cdot 1013 $$\r\nããããã®ã§ïŒ\r\n$$A = \\dfrac{1}{2026} à (2^{2024} + 1012 + 1012 à 1013^{2})$$\r\nãåŸãïŒã«ãŒããäžæãéžã°ãªãå Žåãé€ããªããŠã¯ãããªãããïŒæ±ããã¹ã㯠$A-1$ ã $1009$ ã§å²ã£ãäœãã§ããïŒãã§ã«ããŒã®å°å®çãçšãããªã©ããã°ãã㯠$\\mathbf{668}$ ãšåããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce004/editorial/8808"
}
] | ã$x$ ã«é¢ãã $2024$ 次æ¹çšåŒ
$$ \sum_{k = 0}^{1012} 2024^ {1012-k} x^{2k} = 0 $$
ã¯çžç°ãªã $2024$ åã®è€çŽ æ°è§£ãæã€ã®ã§ïŒãããå
šãŠãèŠçŽ ãšããŠæã€éåã $S$ ãšããŸãïŒãã®ãšãïŒ$S$ ã®**空ã§ãªã**éšåéåã§ãã£ãŠïŒèŠçŽ ãšããŠå«ãŸããè€çŽ æ°ãå
šãŠæãåããããšæŽæ°ãšãªããããªãã®ã®åæ°ãçŽ æ° $1009$ ã§å²ã£ãäœãã解çããŠäžããïŒ |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8146 | E | OMCE004(E) | 700 | 6 | 14 | [
{
"content": "ã$YZ$ ã®äžç¹ã $L^\\prime$ ãšãããšïŒ$AY = AZ$ ãã $\\angle AL^\\prime Y = \\angle AL^\\prime Z = 90^\\circ$ ã§ãã.ããŸãïŒ\r\n$$\\angle YAL^\\prime = \\angle ZAL^\\prime = \\dfrac{\\angle YAZ}{2} = \\angle BAC$$ \r\nã§ããããïŒ\r\n$$\\triangle YAL^\\prime \\sim \\triangle BAH_B, \\quad \\triangle ZAL^\\prime \\sim \\triangle CAH_C$$\r\nãšåããïŒãŸãïŒãã®ãšãïŒ$YA:BA = AL^\\prime:AH_B , \\ \\angle YAB = \\angle L^\\prime AH_B$ ããïŒ$\\triangle YAB \\sim \\triangle L^\\prime AH_B$ ãåããïŒåæ§ã«ïŒ$\\triangle ZAC \\sim \\triangle L^\\prime AH_C$ ãåããïŒããããïŒ\r\n$$\\begin{aligned}\r\nL^\\prime H_B &= YB \\cdot \\dfrac{AH_B}{AB} = XB \\cdot \\cos \\angle BAC \\\\\\\\\r\nL^\\prime H_C &= ZC \\cdot \\dfrac{AH_C}{AC} = XC \\cdot \\cos \\angle BAC\r\n\\end{aligned}$$\r\nãåããïŒãã£ãŠïŒ$$L^\\prime H_B + L^\\prime H_C = XB \\cdot \\cos \\angle BAC + XC \\cdot \\cos \\angle BAC = BC \\cdot \\cos \\angle BAC$$\r\nãåŸãïŒäžæ¹ã§ïŒ$4$ ç¹ $B, H_C, H_B, C$ ãåäžååšäžã«ããããšãã $\\triangle AH_BH_C \\sim \\triangle ABC$ ãåããïŒãã£ãŠïŒ\r\n$$H_BH_C = BC \\cdot \\dfrac{AH_B}{AB} = BC \\cdot \\cos \\angle BAC$$ \r\nãšãªãïŒ$ L^\\prime H_B + L^\\prime H_C = H_BH_C$ ãåŸãïŒãã£ãŠïŒ$L^\\prime$ 㯠$H_BH_C$ äžã«ããïŒ$YZ$ ãš $H_BH_C$ ã®äº€ç¹ã $L^\\prime$ ã§ããããšïŒã€ãŸãïŒ$L^\\prime = L$ ãåŸãïŒãããã $AH$ ãš $YZ$ ã $YZ$ ã®äžç¹ $L = L^\\prime$ ã§äº€ããããïŒ$AH \\perp YZ$ïŒããªãã¡ $YZ \\parallel BC$ ãåŸãïŒãã£ãŠïŒ$\\angle ZYM = \\angle BXM = 90^\\circ - \\angle ABC = \\angle BAH_A$ ãšåããïŒããã§ïŒ\r\n$$\\begin{aligned}\r\n\\angle ZYM &= \\angle AYM - \\angle AYZ = 90^\\circ - \\angle YAM - \\left(90^\\circ - \\dfrac{\\angle YAZ}{2} \\right) \\\\\\\\\r\n&= \\angle BAC - \\angle XAB = \\angle XAC\r\n\\end{aligned}$$\r\nãã $\\angle XAC = \\angle BAH_A$ ãåŸãïŒããããäžè§åœ¢ $ABC$ ã®å€å¿ã¯çŽç· $AX$ äžã«ããã®ã§ïŒçŽç· $AX$ ãšäžè§åœ¢ $ABC$ ã®å€æ¥åã® $A$ ã§ãªãæ¹ã®äº€ç¹ã $D$ ãšãããšïŒ$AD$ ã $\\triangle ABC$ ã®å€æ¥åã®çŽåŸãšãªãïŒããã§ïŒ$\\angle ALY= \\angle AMY = 90^\\circ$ ããïŒ$\\triangle ALM$ ã®å€æ¥å㯠$AY$ ãçŽåŸãšãããšåããïŒãã£ãŠïŒ$\\angle APY = \\angle APD = 90^\\circ$ ãšåããïŒ$3$ ç¹ $Y, P, D$ ãåäžçŽç·äžã«ããããšããã³ $ AP \\perp YD$ ãåããïŒããã« $\\angle XAC = \\angle BAH_{A}$ ããïŒ$\\angle XAB = \\angle CAH_{A} = 45^\\circ$ ãã $\\angle XAM = 45^\\circ$ ããã³ $\\angle AXM = 45^\\circ$ ãåŸãïŒ$\\angle AYM = \\angle AXM = 45^\\circ$ ãåããïŒãã£ãŠïŒ$A, Y, P, M$ ã®å
±åããïŒååšè§ã®å®çãã $\\angle APM = \\angle AYM = 45^\\circ$ ãåŸãïŒããããïŒ$\\triangle APM $ ã«äœåŒŠå®çãé©çšã㊠$AM = \\sqrt{17}$ ãåŸãããïŒãŸãïŒ$\\triangle AMY$ ãš $\\triangle AMX$ ã¯å
±ã« $\\angle M = 90^\\circ$ ã®çŽè§äºç蟺äžè§åœ¢ã®ããïŒ$AY = AX = \\sqrt{34}$ ãåããïŒãã£ãŠïŒ$\\triangle APY$ ã«äžå¹³æ¹ã®å®çãé©çšã㊠$YP = 3$ ãåããïŒããããïŒ$\\triangle APY$ ãš $\\triangle DPA$ ã®çžäŒŒæ¯ã $3 : 5$ ãšåããïŒ$AD = \\dfrac{5\\sqrt{34}}{3}$ ãåŸãããïŒãã£ãŠïŒ$AX = \\sqrt{34}, \\ XD = \\dfrac{2\\sqrt{34}}{3}$ ãšåããïŒããã§ïŒ$\\triangle ABC$ ã®å€æ¥åã®çŽåŸã $AD = \\dfrac{5\\sqrt{34}}{3}$ ã®ããïŒ$\\triangle ABC$ ã«æ£åŒŠå®çãé©çšã㊠$AB = \\dfrac{5\\sqrt{17}}{3}$ ãšåããïŒãã£ãŠïŒ$\\triangle BAX$ ã«äœåŒŠå®çãé©çšã㊠$BX = \\dfrac{\\sqrt{221}}{3}$ ãåããïŒãŸãïŒæ¹ã¹ãã®å®çããïŒ$AX \\cdot XD = BX \\cdot XC$ ã®ããïŒ$XC = \\dfrac{4\\sqrt{221}}{13}$ ãåããïŒ$BC = \\dfrac{25\\sqrt{221}}{39}$ ãåããïŒãã£ãŠïŒ$BC^{2} = \\dfrac{10625}{117}$ ãšãªãïŒç¹ã«è§£çãã¹ãå€ã¯ $\\mathbf{10742}$ ãšãªãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce004/editorial/8146"
},
{
"content": "ãçŽç· $AB, AC$ ã«é¢ã㊠$H_A$ ãšå¯Ÿç§°ãªç¹ãé ã« $H_1, H_2$ ãšããã° $H_2, H_B, L, H_C, H_1$ ã¯åäžçŽç·äžã§ïŒäžè§åœ¢ $AYZ$ ãšäžè§åœ¢ $AH_1H_2$ ãçžäŒŒãã $4$ ç¹ $A, Y, H_1, L$ ããã³ $A, L, Z, H_2$ ã¯ããããåäžååšäžïŒããã§äžè§åœ¢ $AH_AX, AH_1Y, AH_2Z$ ã¯å
šãŠååãªçŽè§äžè§åœ¢ãªã®ã§ $\\angle ALY=\\angle AH_1Y=\\angle AH_AX=\\angle AH_2Z=\\angle ALZ=90\\degree$ ãã $\\angle YAL=\\angle ZAL=\\angle BAC$ ãªã®ã§ $\\angle BAX= \\angle BAY=\\angle CAH$ ããäžè§åœ¢ $ABC$ ã®å€å¿ $O$ ã¯çŽç· $AX$ äžã«ããïŒ\\\r\nãããšã¯å
¬åŒè§£èª¬ãšåæ§ã§ãããïŒç¹ $O$ ã«é¢ããŠ ç¹ $A$ ãšå¯Ÿç§°ãªç¹ã$D$ïŒç¹ $M$ ã«é¢ããŠç¹ $P$ ãšå¯Ÿç§°ãªç¹ã $Q$ ãšããã°ïŒ\r\n$$BC=DP\\times\\dfrac{AM}{AQ}=\\dfrac{AP^2}{YP}\\times\\dfrac{AM}{AQ}$$\r\n$$YP=AP-PM\\times\\sqrt2$$\r\nããèšããã®ã§ïŒãããããæ±ããããšã¯ã§ããïŒãïŒããã瀺ãåŽåãããå
¬åŒè§£èª¬ã®èšç®ã®æ¹ã軜ããšæãããã®ã§éæšå¥šïŒ",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce004/editorial/8146/538"
}
] | ãåå¿ã $H$ ã§ããäžè§åœ¢ $ABC$ ãããïŒ$A$, $B$, $C$ ããçŽç· $BC$, $CA$, $AB$ ã«äžãããåç·ã®è¶³ããããã $H_A, H_B, H_C$ ãšããŸãïŒç·å $H_AC$ äžïŒäž¡ç«¯ç¹ãé€ãïŒã«ç¹ $X$ ãåãïŒçŽç· $AB, AC$ ã«é¢ã㊠$X$ ãšå¯Ÿç§°ãªç¹ããããã $Y, Z$ ãšãããšïŒ$3$ çŽç· $YZ , AH , H_BH_C$ ã $1$ ç¹ $L$ ã§äº€ãããŸããïŒããã«ç·å $XY$ ã®äžç¹ã $M$ ãšãïŒäžè§åœ¢ $ALM$ ã®å€æ¥åãšäžè§åœ¢ $ABC$ ã®å€æ¥åã® $A$ ã§ãªãæ¹ã®äº€ç¹ã $P$ ãšãããšïŒ
$$ AP = 5, \quad PM = \sqrt{2}, \quad \angle ACB = 45^\circ $$
ãšãªããŸããïŒãã®ãšãïŒèŸº $BC$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\sqrt\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠäžãã. |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8673 | F | OMCE004(F) | 700 | 12 | 38 | [
{
"content": "ã以äžïŒããŒãåºãã人ã $P$ïŒã°ãŒãåºãã人ã $G$ïŒãã§ããåºãã人ã $C$ ãšè¡šãïŒãŸãïŒäžè¬æ§ã倱ããã«ïŒæ®ã $3$ 人ã«ãªã£ã段é㧠$P, G, C$ ããã®é ã«æèšåãã«äžŠãã§ãããšããïŒãŸãïŒ$k$ åæŠãçµãã£ãçŽåŸã« $P$ ãš $G$ ããã®é ã«æèšåãã«ãªãããã«é£ãåã£ãŠãããšãïŒãã®äºäººã®éã«ã㊠$k$ åæŠã§è±èœãã人ã«ã€ããŠèããïŒ$P$ ã®æèšåãã«é£ã«ãã人ã§è±èœãã人ããããšãããšïŒãã®äººã¯äž¡é£ã®ã©ã¡ãã«ãåã£ãŠããªãããïŒ$P$ ã $G$ ã ãšåããïŒãŸãïŒ$P$ ã ãšãããšïŒãã®æèšåãã«é£ã®äººã $C$ ãšç¢ºå®ãïŒãã®äººã $k$ åæŠã§è±èœããŠãããšåãããïŒããã¯ãã®äººã $P$ ã«åå©ãåŸç¹ã $0$ ç¹ä»¥äžã§ããããšã«ççŸããïŒãã£ãŠïŒ$P$ ã®æèšåãã«é£ã®äººã¯ $G$ ããããããªããšåããïŒãŸãïŒãã®æèšåãã«é£ã«è±èœãã人ãç¶ããŠãããšãããšïŒãã㯠$P, \\dots$ ã $G, P, \\dots$ ã®ã©ã¡ããã®ããã«ç¶ãããšãšãªãïŒãããã®å Žåã $P$ ãè±èœããã«ççŸããïŒãã£ãŠïŒè±èœãã人㯠$G$ äžäººã誰ãããªããã®ã©ã¡ããã§ããïŒ$P, G$ ã®é çªãéã®å Žåãåæ§ãšãªãïŒé£ãåãäºäººã $G, C$ ã®å Žåã¯è±èœãã人ã $C$ äžäººã誰ãããªãããšïŒåã³é£ãåãäºäººã $C, P$ ã®å Žåã¯è±èœãã人ã $P$ äžäººã誰ãããªãããšãåæ§ã«ããŠåããïŒ\\\r\nã次ã«ïŒ$k$ åæŠãçµãã£ãçŽåŸã« $P$ ãš $P$ ããã®é ã«æèšåãã«ãªãããã«é£ãåã£ãŠãããšãïŒãã®äºäººã®éã«ã㊠$k$ åæŠã§è±èœãã人ã«ã€ããŠèããïŒ$P$ ã®æèšåãã«é£ã«ãã人ã§è±èœãã人ããããšãããšïŒãã®äººã¯$P$ ã«åã£ãŠããªãããïŒ$P$ ã $G$ ã§ããïŒãŸãïŒ$P$ ã ãšãããšïŒãã®æèšåãã«é£ã®äººã $C$ ãšãªãïŒãã®äººãè±èœããªãããççŸããïŒãã£ãŠïŒ$G$ ãšç¢ºå®ããïŒãŸãïŒ$G$ ã®æèšåãã«é£ã®äººã§è±èœãã人ããããšãããšïŒãã®äººã¯ $P$ ã $G$ ãšãªããïŒ$P$ ã ãšè±èœããççŸããããïŒ$G$ ã«ç¢ºå®ããïŒããã«é£ã«è±èœãã人ããããšãããšïŒãã㯠$P$ ãããªããïŒãã¯ã $P$ ã¯è±èœããççŸããïŒããããïŒè±èœãã人㯠$G$ äºäººã $G$ äžäººã誰ãããªããã®ããããã§ããïŒ$G$ ãš $G$ïŒ$C$ ãš $C$ ããã®é ã«æèšåãã«ãªãããã«é£ãåã£ãŠããå Žåãåæ§ã«èããããšãã§ããïŒ\\\r\nãããã§ïŒ$k$ åæŠãçµãã£ãçŽåŸã«åã«æ®ã£ãŠããäœäººã«ã€ããŠïŒåºããæãç°ãªããããªé£ãåãäºäººã®çµã®æ°ã $a_{n-k}$ïŒåºããæãåããããªé£ãåãäºäººã®çµã®æ°ã $b_{n-k}$ ãšãã (ãã ãïŒäºäººã®é çªã¯åºå¥ããªããã®ãšãã)ïŒ ãã®ãšãïŒ$a_{0} = 3, b_{0} = 0$ ãšåããïŒãŸãïŒ$a_{n}, b_{n}$ ã¯ãããã $1200$ 人å
šå¡ãåã«æ®ã£ãŠããæåã®ç¶æ
ã«ãããå€ãè¡šããã®ãšããïŒãã®ãšãïŒäžèšã®è°è«ããïŒ\r\n$$a_{m+1} \\leq a_{m} + 2b_{m}, \\ b_{m+1} \\leq b_{m} + a_{m}$$\r\nãåãã $(0 \\leq m \\leq n-1)$ïŒãŸãïŒ$k$ åæŠãçµãã£ãçŽåŸã«åã«æ®ã£ãŠããäœäººã®æ°ã¯ $a_{n-k} + b_{n-k}$ ãšãªãããšã容æã«åããïŒããããïŒ\r\n$$a_{1} \\leq 3, \\ b_{1} \\leq 3 , \\dots , a_{7} \\leq 717, \\ b_{7} \\leq 507$$\r\nãåããïŒãããã $n \\geq 7$ ãåŸãïŒãŸãïŒ$n = 7$ ãå®çŸããåæç¶æ
ãååšããããšã¯å®¹æã«åããããïŒ$n$ ã®æå°å€ã $7$ ãšåããïŒ\\\r\nã以äžïŒ$n = 7$ ãšããŠèããïŒ$a_{6} + b_{6} \\leq 507$ ããïŒäžåæŠã§è±èœãã人æ°ã¯ $693$ 人以äžãšåããïŒãŸãïŒäžåæŠã§æ£ã®åŸç¹ãåŸã人ã®ç¹æ°ã®åèšã¯ $a_{7} \\leq 717$ ãšãåããïŒå
šå¡ã®åŸç¹ã®åèšã $0$ ã§ããããšïŒåã³è±èœãã人ã®åŸç¹ã¯ $-1$ ãŸã㯠$-2$ ã§ããããšãèžãŸãããšïŒ$B \\leq 717 - 693 = 24$ ãåããïŒãŸãïŒ$2$ åæŠã§è±èœãã人ã«ã€ããŠèãããšïŒãã®äººã¯ $2$ åæŠã§ $-1$ 以äžã®åŸç¹ãåã£ãããšã«ãªãïŒãã®ãããªäººã¯ $1$ åæŠã§ $1$ ç¹ä»¥äžã®åŸç¹ããåŸãããªãïŒãŸãïŒ$1$ åæŠã§è±èœãã人ã¯åœç¶ïŒ$1$ åæŠã§$-1$ 以äžã®åŸç¹ããåŸãŠããªãããïŒçµå± $1$ åæŠã§ $2$ ç¹ãåã£ã人㯠$2$ åæŠåŸã«åã«æ®ã£ãŠãããšåããïŒãã£ãŠïŒ$A \\leq a_{5} + b_{5} \\leq 123 + 87 = 210$ ãšåããïŒ\\\r\nã以äžïŒãããã®çå·ãå®çŸããåæç¶æ
ãååšããããšã瀺ãïŒãŸãïŒ\r\n$$a_{0} = 3, \\ b_{0} = 0, \\ a_{m+1} = a_{m} + 2b_{m}, \\ b_{m+1} = b_{m} + a_{m} \\ \\ \\ (0 \\leq m \\leq n-1)$$\r\nããæºããããã«é·ç§»ããå ŽåãèããïŒãããšïŒ$a_{7} + b_{7} = 1224$ ãšãªãïŒå®éã®äœäººã®æ°ãã $24$ å€ããªã£ãŠããŸãïŒããã§ïŒåæç¶æ
ã«ãã㊠$P, G, G, P$ ãŸã㯠$G, C, C, G$ ãŸã㯠$C, P, P, C$ ãšé£ç¶ããŠäžŠãã§ããéšåãèããïŒ(ãã®ãããªéšåã¯å°ãªããšã $b_{6} = 210$ ç®æã¯ããïŒ) ããã $P, G, P$ ã®ããã«éã® $2$ ã€ã $1$ ã€ã«çœ®ãæããæäœã $24$ ç®æã«è¡ãããšã§ïŒå®éã®äœäººã®æ°ãšäžèŽããïŒãŸãïŒããã«ããïŒ$1$ åæŠçµäºä»¥éã®ç¶æ
ã¯å€åããïŒãŸãïŒ$B = 24$ïŒåã³ $A = a_{5} + b_{5} = 210$ ãå®çŸããŠããããšã容æã«åããïŒ\\\r\nã以äžããïŒè§£çãã¹ãå€ã¯ $210 \\times 24 = \\mathbf{5040}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce004/editorial/8673"
}
] | ãOMC æã«ã¯ $1200$ 人ã®äœäººãããŸãïŒãããã®äœäººå
šå¡ã§ä»¥äžã®ã«ãŒã«ã«åŸãããããã倧äŒãè¡ãããšã«ããŸããïŒ
- ãŸããå
šå¡ãã©ã®æãåºããäºåã«æ±ºããŠããïŒãã®æã¯å€§äŒãçµäºãããŸã§å€ããããšã¯ãªãïŒ
- äžåæŠã¯ $1200$ 人å
šå¡ã§è¡ãïŒå
šå¡ãå圢ã«äžŠã³ïŒäºåã«æ±ºããŠãããæãããšã«å·Šé£ã®äœäººïŒå³é£ã®äœäººã®ãããããšããããããè¡ãïŒ
- ããããã®åè² ã«ãããŠïŒåã£ãå Žå㯠$1$ ç¹ãïŒè² ããå Žå㯠$-1$ ç¹ãç²åŸãïŒãããã®å Žåã¯åŸç¹ã¯ç²åŸããªãïŒ(ã€ãŸãïŒåäœäººã®åŸç¹ã¯ $-2$ ç¹ä»¥äž $2$ ç¹ä»¥äžã®æŽæ°å€ãåãåŸãïŒ)
- å
šãŠã®ããããããçµäºããã®ã¡ïŒåŸç¹ã $0$ ç¹**æªæº**ã®äœäººã¯äžæã«åããæãïŒãã®åŸæ®ã£ããã¹ãŠã®äœäººã®åŸç¹ã $0$ ã«æ»ãïŒ
- äºåæŠä»¥éã¯ïŒåã«æ®ã£ãäœäººã®ã¿ã§äžåæŠãšåæ§ã®ã«ãŒã«ã§è¡ãïŒ
ã
ãã®ãšãïŒ$n$ åæŠãçµãã£ãçŽåŸã«åã«æ®ã£ãäœäººã¯ $3$ 人ãšãªãïŒããŒïŒã°ãŒïŒãã§ããåºãã人ã $1$ 人ãã€ããŸããïŒãã®ãšãïŒ$n$ ãšããŠèããããæå°å€ãå®çŸãããããªå ŽåãèãããšãïŒäžåæŠã§ $2$ ç¹ãç²åŸãã人æ°ãšããŠããåŸãæ倧å€ã $A$ïŒäžåæŠã§ $-2$ ç¹ãç²åŸãã人æ°ãšããŠããåŸãæ倧å€ã $B$ ãšå®ããã®ã§ïŒ$A \times B$ ã解çããŠãã ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/7601 | A | OMCB012(A) | 100 | 358 | 359 | [
{
"content": "ãå¹³æ¹æ°ãæ£æŽæ° $n$ ãçšã㊠$n^2$ ãšè¡šããšïŒ$n^2-1=(n+1)(n-1)$ ãçŽ æ°ã§ãããã $n-1=1$ïŒåŸã£ãŠåé¡ã®æ¡ä»¶ã«åœãŠã¯ãŸãå¹³æ¹æ°ã¯ $n^2=\\mathbf{4}$ ã®ã¿ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/7601"
}
] | ãçŽ æ°ã« $1$ ã足ããŠåŸãããå¹³æ¹æ°ãšããŠãããããã®ã®ç·åãæ±ããŠãã ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/8383 | B | OMCB012(B) | 100 | 344 | 347 | [
{
"content": "ã$xy-2x-3y+6=(x-3)(y-2)$ ã§ããïŒ$x-3,y-2$ ã¯ã©ã¡ããæ¡ä»¶äžã«ãããŠåžžã«éè² ã§ããããïŒæ倧å€ã¯ $(x,y)=(19,21)$ ã®ãšã㧠$304$ïŒæå°å€ã¯ $(x,y)=(16,t)$ ã®ãšã㧠$13t-26$ ã§ããïŒãã®å·®ã¯ $330-13t$ ã§ããããïŒ$330-13t=135$ ã解ã㊠$t=\\mathbf{15}$ ãåŸãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/8383"
}
] | ãå®æ° $t$ 㯠$2\leq t\leq 21$ ããã³æ¬¡ãã¿ãããŸããïŒ$t$ ã®å€ã解çããŠãã ããïŒ
- å®æ° $x,y$ ã $16\leq x\leq 19,t\leq y\leq 21$ ãæºããããã«åããšãïŒ
$$xy-2x-3y+6$$
ã®æ倧å€ãšæå°å€ã®å·®ã¯ $135$ ã§ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/3194 | C | OMCB012(C) | 200 | 289 | 320 | [
{
"content": "ãäžè¬æ§ã倱ãã $x \\leq y \\leq z$ ãšããŠèãããš $x^3+3x\\leq 33$ ã§ããïŒãããã $x=1,2$ ã§ããïŒ\\\r\nã$x=1$ ã®ãšãïŒ$yz+y+z=32$ ãã\r\n$$(y+1)(z+1)=33$$\r\nã§ããïŒ$y+1\\geq 2$ ãã $y+1=3,z+1=11$ ãšãªãã»ããªãïŒãã£ãŠ $(x,y,z)=(1,2,10)$ïŒ\\\r\n ã$x=2$ ã®ãšãïŒ$2yz+y+z=31$ ãã\r\n$$(2y+1)(2z+1)=63$$\r\nã§ããïŒ$2y+1\\geq 2x+1=5$ ãã\r\n$(2y+1,2z+1)=(7,9)$ ãšãªãã»ããªãïŒãã£ãŠ $(x,y,z)=(2,3,4)$ïŒ\\\r\nã$x \\leq y \\leq z$ ã®ä»®å®ãå€ããšïŒãããã $3!$ éãããã®ã§æ±ããç·åã¯\r\n$$\\\\{(1+2+10)+(2+3+4)\\\\}Ã3!=\\textbf{132}.$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/3194"
}
] | ã$xyz+x+y+z=33$ ãã¿ããæ£ã®æŽæ°ã®çµ $(x,y,z)$ ãã¹ãŠã«ã€ããŠïŒ
$x+y+z$ ã®ç·åãçããŠãã ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/6768 | D | OMCB012(D) | 200 | 221 | 274 | [
{
"content": "ã$f$ ãå¶æ°æ¬¡ã®é
ãããªãããšãã $f$ ã¯å¶é¢æ°ãªã®ã§, $f(-1)=f(-2)=f(-3)=f(-4)=5$ ã§ãã. åŸã£ãŠ, \r\n$$f(x)=(x-4)(x-3)(x-2)(x-1)(x+1)(x+2)(x+3)(x+4)+5$$\r\nãåãããã, $f(5)=\\mathbf{72581}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/6768"
},
{
"content": "ãæ¡ä»¶ãã $f(x)=x^8+ax^6+bx^4+cx^2+d$ ãšè¡šããããšãåãããŸãïŒãã®ãšã $t=x^2$ ãšãããš\r\n$$f(x)=\\left(x^4\\right)^2+a\\left(x^3\\right)^2+b\\left(x^2\\right)^2+c\\left(x^2\\right)+d=t^4+at^3+bt^2+ct+d$$\r\nãšããããã«èŠãããšãã§ããŸãïŒäžããããŠãã $f(i)$ ã®å€ããã¹ãŠ $5$ ã§ããããšããïŒ$f$ ãã $5$ ãåŒããšå æ°å®çã䜿ãã圢ã«ã§ãããã§ãïŒ$g(t)=(t^4+at^3+bt^2+ct+d)-5$ ãšãããšïŒæ¡ä»¶ãã $g(1)=g(4)=g(9)=g(16)=0$ ã§ããã®ã§ïŒå æ°å®çãã\r\n$$g(t)=(t-1)(t-4)(t-9)(t-16)$$\r\nãšè¡šããŸãïŒ$x^8$ ã®ä¿æ°ïŒããªãã¡ $t^4$ ã®ä¿æ°ã $1$ ã§ããããšã«æ³šæïŒïŒïŒæ±ããããã®ïŒã«è¿ããã®ïŒã¯ $g(25)$ ã§ããïŒãã®å€ã¯\r\n$$24\\times 21\\times 16\\times 9=72576$$\r\nã§ãïŒä»¥äžãã $f(5)=g(25)+5=\\boxed{72581}$ ãšæ±ãŸããŸããïŒ",
"text": "眮æããŠã¿ããïŒfor beginnersïŒ",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/6768/534"
}
] | ã以äžãå
šãŠæºãã $8$ 次ã®å®æ°ä¿æ°å€é
åŒ $f$ ã¯äžæã«ååšããã®ã§ïŒ$f(5)$ ã解çããŠãã ãã.
- å¥æ°æ¬¡ã®é
ã®ä¿æ°ã¯å
šãŠ $0$ ã§ãã.
- $8$ 次ã®é
ã®ä¿æ°ã¯ $1$ ã§ãã.
- $f(1)=f(2)=f(3)=f(4)=5$. |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/3865 | E | OMCB012(E) | 200 | 243 | 288 | [
{
"content": "ãåæ¡ã®æ°ã $a\\lt b$ ã® $2$ çš®é¡ãããªãè¯ãæ°ã¯ ${}_5 \\mathrm{C}_1+{}_5 \\mathrm{C}_2+{}_5 \\mathrm{C}_3+{}_5 \\mathrm{C}_4=30$ åããïŒ$2^5-2$ ãšèããŠãããïŒïŒãã㧠$\\overline{aaaab}$ ã«ã¯ $\\overline{bbbba}$ ãïŒ$\\overline{aaabb}$ ã«ã¯ $\\overline{bbbaa}$ ã察å¿ãããèŠé 㧠$2$ ã€ãã€ãã¢ã«ããããšã§ïŒ$30$ åã®ç·å㯠$11111(a+b)\\times 15$ ã§ããïŒ\\\r\nããã㧠$(a,b)$ ãšããŠããåŸããã®ãã¹ãŠ $(1,2),(1,3),\\ldots,(8,9)$ ã«ãããŠïŒ$1$ ãã $9$ ã¯ãããã $8$ åãã€çŸããããïŒ$a+b$ ã®ç·å㯠$45\\times 8$ ã§ããïŒä»¥äžããæ±ããç·å㯠$\\textbf{59999400}$ ãšèšç®ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/3865"
},
{
"content": "ãïŒåïŒïŒïŒå¹³åïŒÃïŒåæ°ïŒãçšããæ¹æ³ã玹ä»ããïŒ\\\r\nã察称æ§ãããïŒå¹³åã容æã«æ±ãŸãåé¡ã«ã€ããŠã¯ïŒéåžžã«äœ¿ãããããã¯ããã¯ã§ããïŒ\r\n\r\n---\r\n\r\nã$\\overline{abcde}$ ãè¯ãæ°ã ãšãããšïŒåäœã®æ°ã $10-a,10-b,\\cdots$ ã«çœ®ãæããæ°ãè¯ãæ°ã§ããïŒå
·äœäŸãšã㊠$26226$ ãš $84884$ïŒïŒãã®ãã㪠$2$ æ°ã®çµã«ã€ããŠïŒãããã®æ°ã®å¹³åã¯å¿
ã $55555$ ã«ãªãïŒãã£ãŠïŒè¯ãæ°ã®å¹³åã $55555$ ã§ããïŒ\\\r\nãããšã¯ïŒè¯ãæ°ã®åæ°ãæ±ããã°ããïŒè¯ãæ°ã«äœ¿ããã $2$ çš®é¡ã®æ°ã®éžã³æ¹ã ${}\\_{9}\\mathrm{C}\\_{2}$ éãããïŒããããã®å Žåã«ã€ããŠïŒ$30$ éãã®æ°ãäœãããšãã§ããïŒ\\\r\nããã£ãŠæ±ããã¹ãå€ã¯ $55555Ã({}\\_{9}\\mathrm{C}\\_{2}Ã30)=\\mathbf{59999400}$\r\n\r\n---\r\n\r\nã以äžã¯äœè«ïŒ\\\r\nã$n$ 以äžã®æ£æŽæ°ã§ $n$ ãšäºãã«çŽ ãªæ°ã®å㯠$\\dfrac{n}{2}\\varphi(n)$ ãšãªããïŒããã¯ïŒåïŒïŒïŒå¹³åïŒÃïŒåæ°ïŒã®å
·äœäŸã ãšèšããïŒ\\\r\nãå¥ã®äŸãšããŠïŒçå·®æ°åã®åã®å
¬åŒãïŒ$\\frac{1}{2}Ã(åé
+æ«é
)=(å¹³å)$ ã§ããããšã«æ³šç®ããã°ïŒïŒåïŒïŒïŒå¹³åïŒÃïŒåæ°ïŒã®å
·äœäŸã ãšèšããïŒ\r\n\r\nãé¡é¡ã解ãããæ¹ã¯ä»¥äžãã©ããïŒ\\\r\nãã[47C](https:\\/\\/onlinemathcontest.com\\/contests\\/omc047\\/tasks\\/1921)ïŒ[84F](https:\\/\\/onlinemathcontest.com\\/contests\\/omc084\\/tasks\\/2169)ïŒ[95E](https:\\/\\/onlinemathcontest.com\\/contests\\/omc095\\/tasks\\/272)",
"text": "å¹³åïŒæåŸ
å€ïŒã䜿ã",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/3865/530"
}
] | ãåé²æ³è¡šèšã§åæ¡ã®æ°ã $0$ 以å€ã®ã¡ããã© $2$ çš®é¡ã®æ°ãããªãæ°ã**è¯ãæ°**ãšåŒã³ãŸãïŒäŸãã° $377$ ã $9494$ ã¯è¯ãæ°ã§ããïŒ$888$ ã $2022$ ã¯è¯ãæ°ã§ã¯ãããŸããïŒã¡ããã© $5$ æ¡ã®è¯ãæ°ã®ç·åãæ±ããŠãã ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/8432 | F | OMCB012(F) | 200 | 149 | 206 | [
{
"content": "ã$A$ ãã $BC$ ã«äžãããåç·ã®è¶³ã $J$ ãšãããšïŒ$BP = PJ$ ããã³å¹³è¡ç·ã®æ¯ã®æ§è³ªãã\r\n$$AE:EC=JQ:QC=1:5$$\r\nã§ããïŒãŸã $B$ ãã $AC$ ã«äžãããåç·ã®è¶³ã $K$ ãšãããšïŒ$AE = EK$ ãã\r\n$$AE:EK:KC=1:1:4$$\r\nãåŸãïŒ$â³AJC \\sim â³BKC$ ãªã®ã§\r\n$$AC:6 = AC : JC=BC:KC=10:\\dfrac{2}{3}AC$$ ãã $AC^2=\\bf{90}$ \r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/8432"
}
] | ãéè§äžè§åœ¢ $ABC$ ã«ãããŠèŸº $AB$ ã®äžç¹ã $D$ ãšãïŒ$D$ ããçŽç· $AC$ ã«ããããåç·ã®è¶³ã $E$ ãšããŸãïŒ$D,E$ ããçŽç· $BC$ ã«ããããåç·ã®è¶³ããããã $P,Q$ ãšãããšãïŒ$$BP=2, \quad PQ=3, \quad QC=5$$
ãæãç«ã¡ãŸããïŒ$AC^2$ ãæ±ããŠãã ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/7177 | G | OMCB012(G) | 300 | 125 | 205 | [
{
"content": "ãéšå¡æ°ã $555, 666, 777, 567$ 人ã®éšæŽ»ããããã $A, B, C, D$ ã§è¡šãïŒã¡ããã© $2$ ã€ã®éšæŽ»ã«å
¥ã£ãŠããã®ã $x$ 人ã§ãããšãïŒãã® $x$ 人ã®ãã¡åéšæŽ»ã«å
¥ã£ãŠããã®ã $a,b,c,d$ 人ãããšããïŒ$a+b+c+d=2x$ ã§ããããšã«æ³šæããïŒããã $x$ 人以å€ã®ååžã«ã€ããŠæ³šç®ããããšã§ïŒã\r\n$$\\max(555-a, 666 - b, 777 - c, 567 - d) \\leq 1000 - x$$\r\nãæãç«ã€ããšãåããïŒããããç¹ã«\r\n$$2565-2x = (555-a)+(666-b)+(777-c)+(567-d) \\leq 4(1000-x)$$\r\nãåŸãããïŒ$x\\leq 717$ ãå¿
èŠã§ããïŒ\\\r\nãéã« $x=717$ ã®ãšãïŒããšãã° $a = 272, b= 383 , c= 494, d = 285$ ãšãïŒ$49$ 人ã $A, C$ ã«ïŒ $160$ 人ã $B, C$ ã«ïŒ$285$ 人ã $C,D$ ã«ïŒ$223$ 人ã $A, B$ ã«ïŒ$1$ 人ã $A,B,C$ ã«ïŒ$282$ 人ã $A,B,C,D$ ã«å
¥ãããšã§æ§æå¯èœã§ããïŒãã£ãŠçã㯠$\\mathbf{717}$ 人ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/7177"
},
{
"content": "ãã¡ããã© $i$ åã®éšæŽ»ã«å
¥ã£ãŠããçåŸã®æ°ã $a_i$ ãšçœ®ãïŒ$a_i$ ã¯å
šãŠéè² æŽæ°ã§ããïŒïŒ\\\r\nã$1000$ 人ã®çåŸãåšç±ããŠããããšã¯ $a_0+a_1+a_2+a_3+a_4=1000$ ãšæžãæãããïŒ$4$ ã€ã®éšæŽ»ã®ããããã®éšå¡æ°ã®æ¡ä»¶ã¯æ¬¡ã®ããã«æžãæããããïŒ\r\n$$\\tag{1} a_1+2a_2+3a_3+4a_4=555+666+777+567=2565$$\r\nãããã§ïŒã€ãããç®ã®ãããªçºæ³ã䜿ããïŒ\\\r\nã$(a_0, a_1, a_2, a_3, a_4)=(0,0,1000,0,0)$ ã§ããã°ïŒ$a_1+2a_2+3a_3+4a_4=2000$ ã§ããïŒ\\\r\nããããã $a_2$ ã $1$ æžãããŠïŒ $a_3$ ã $1$ å¢ããã°ïŒå³èŸºã®å€ã¯ $1$ å¢ããïŒäžæ¹ $a_2$ ã $1$ æžãããŠïŒ $a_4$ ã $1$ å¢ããã°ïŒå³èŸºã®å€ã¯ $2$ å¢ããïŒãŸãïŒãããã®æäœã¯ãããã $a_0+a_1+a_2+a_3+a_4=1000$ ãä¿ã£ãŠããïŒ\\\r\nãåŒ $(1)$ ãæºãã $a_2$ ã®æ倧å€ãæ±ãããã£ãã®ã§ïŒäžèšã®æäœãã§ããã ãå°ãªãåæ°ã§ïŒå³èŸºã $2565$ ã«ããã°ããïŒãã®ããã«ã¯ïŒ$a_2 \\rightarrow a_4$ ã®æäœã $282$ åïŒ$a_2 \\rightarrow a_3$ ã®æäœã $1$ åããã°ããïŒãã®ãšã $a_2=717$ ã§åŒ $(1)$ ãéæã§ããïŒ\r\n\r\n---\r\n\r\nã以äžã®è°è«ã¯ïŒæ£ç¢ºã«ã¯ $a_2 \\leq 717$ ã瀺ããã ãã§ããïŒå
¬åŒè§£èª¬ã®ããã«æ§æããªããã° $a_2=717$ ã瀺ããããšã«ã¯ãªããªãïŒ",
"text": "ã€ãããç®ã®ãããªçºæ³",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/7177/531"
}
] | ãOMCåŠåã«ã¯ $1000$ 人ã®çåŸãåšç±ããŠããŸãïŒOMCåŠåã«ã¯ $4$ ã€ã®éšæŽ»ãããïŒéšå¡æ°ã¯ãããã $555$ 人ïŒ$666$ 人ïŒ$777$ 人ïŒ$567$ 人ã§ãïŒã¡ããã© $2$ ã€ã®éšæŽ»ã«æå±ããŠããçåŸã®äººæ°ãšããŠããããæ倧å€ã解çããŠãã ããïŒãã ãïŒã©ã®éšæŽ»ã«ãæå±ããŠããªãçåŸãïŒ$3$ ã€ä»¥äžã®éšæŽ»ã«æå±ããŠããçåŸãããŠãããŸããŸããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/4575 | H | OMCB012(H) | 300 | 52 | 85 | [
{
"content": "ãåããåã $C$ ,ãã®äžå¿ã $P$ ãšãïŒ$A(0,1),B(0,-1)$ ãšããïŒãŸãïŒ$A,B$ ãããããäžå¿ãšããååŸ $2$ ã®åããããã $C_1,C_2$ ãšããïŒãã®ãšãïŒ$PA=PB=\\dfrac{1}{\\cos \\alpha}$ ãšãªãïŒäžå¿ã®è·é¢ãšååŸã®é¢ä¿ããå $C$ ã¯å $C_1$ ãšå $C_2$ ã«å
æ¥ããªããåãããšããããïŒããããïŒæ±ããééé åã¯ïŒ$x\\geq 0$ ã®ç¯å²ã§ã¯å $C_1$ ãšå $C_2$ ã®å
±ééšåã§ããïŒ$x\\leq 0$ ã®ç¯å²ã§ã¯ $\\alpha=0$ ã®ãšãã®å $C$ ã®ååã§ããããšããããïŒãã£ãŠïŒééé åã®é¢ç©ã¯ä»¥äžã®å€ã«ãªãïŒ\r\n$$\\Bigl(\\dfrac{4\\pi}{3}-\\sqrt{3}\\Bigr)+\\dfrac{\\pi}{2}=\\dfrac{11\\pi}{6}-\\sqrt{3}$$\r\nç¹ã«è§£çãã¹ãå€ã¯ $\\mathbf{20}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/4575"
},
{
"content": "ãåè:[é«æ ¡æ°åŠã®çŸããç©èª](https:\\/\\/manabitimes.jp\\/math\\/1178) \\\r\nãçè
ã®å匷äžè¶³ã«ã€ã, è¥å¹²å³å¯æ§ãæ¬ ããšæããŸã.\r\n---\r\nãæ²ç·ã®æ¹çšåŒã¯å®æ° $\\alpha$ ãçšããŠ, \r\n$$f(x,y,\\alpha)=(x-\\tan \\alpha)^2+y^2-(2-\\frac{1}{\\cos \\alpha})^2=0$$\r\n ãšè¡šããã. ãã®ãšã, \r\n$$\\frac{d}{d\\alpha}f(x,y,\\alpha)=-\\frac{2}{\\cos^2\\alpha}(x-2\\sin \\alpha)=0$$\r\nãšãªã, $f(x,y,\\alpha)=0$ ãš $\\dfrac{d}{d\\alpha}f(x,y,\\alpha)=0$ ãé£ç«ã㊠$\\alpha$ ãæ¶å»ããŠåŸãããæ¹çšåŒã, $f(x,y,\\alpha)=0$ ã®å
絡ç·(ä»»æã® $\\alpha$ ã«å¯ŸããŠ, $f(x,y,\\alpha)=0$ ã«æ¥ããæ²ç·)ãšãªãããšãç¥ãããŠãã.\\\r\n ã$\\dfrac{d}{d\\alpha}f(x,y,\\alpha)=0$ ãã, $\\sin \\alpha=\\dfrac{1}{2}x$ ã $f(x,y,\\alpha)=0$ ã«ä»£å
¥ããŠã¿ã. ($\\cos \\alpha=0$ ã®ãšãæ²ç·ãå®çŸ©ãããªãã®ã§, 以éã $\\cos \\alpha\\neq 0(\\iff \\sin^2\\alpha-1\\neq 0)$ ãšããŠåŒå€åœ¢ãã.)\r\n$$\r\n\\begin{aligned}\r\n&(x-\\tan \\alpha)^2+y^2-(2-\\frac{1}{\\cos \\alpha})^2=0\\\\\\\\\r\n\\iff & x^2-2\\tan\\alpha x+y^2+\\frac{4}{\\cos\\alpha} -4+\\tan^2\\alpha-\\frac{1}{\\cos^2\\alpha}=0\\\\\\\\\r\n\\iff &x^2+y^2-5=\\frac{2\\sin \\alpha x-4}{\\cos \\alpha}\\\\\\\\\r\n\\implies &(x^2+y^2-5)^2=\\frac{(2\\sin \\alpha x-4)^2}{1-\\sin^2 \\alpha}\\\\\\\\\r\n\\iff &(x^2+y^2-5)^2=\\frac{(x^2-4)^2}{1-(\\frac{x}{2})^2}\\\\\\\\\r\n\\iff &(x^2+y^2-5)^2+4(x^2-4)=0\\\\\\\\\r\n\\iff &x^4+y^4+2x^2y^2-10x^2-10y^2+25+4x^2-16=0\\\\\\\\\r\n\\iff &x^4+(2y^2-6)x^2+(y^4-10y^2+9)=0\\\\\\\\\r\n\\iff &x^4+(2y^2-6)x^2+(y^2-9)(y^2-1)=0\\\\\\\\\r\n\\iff &x^4+(2y^2-6)x^2+(y-3)(y+1)(y+3)(y-1)=0\\\\\\\\\r\n\\iff &x^4+(2y^2-6)x^2+(y^2-2y-3)(y^2+2y-3)=0\\\\\\\\\r\n\\iff &(x^2+y^2-2y-3)(x^2+y^2+2y-3)=0\\\\\\\\\r\n\\iff &(x^2+(y-1)^2-4)(x^2+(y+1)^2-4)=0\\\\\\\\\r\n\\end{aligned}\r\n$$\r\nãã£ãŠ, $f(x,y,\\alpha)=0$ ã®å
絡ç·ã¯ $(0,1)$ ãäžå¿ãšããååŸ $2$ ã®åãš $(0,-1)$ ãäžå¿ãšããååŸ $2$ ã®åã§ããããšãèšç®ã«ããåŸããã.",
"text": "å
絡ç·ã®è©±",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/4575/522"
},
{
"content": "ã$\\tan \\alpha =a$ ãšãããš $\\dfrac{1}{\\cos \\alpha}=\\sqrt{a^2+1}$ ã§ããïŒãã® $a$ ãçšããŠïŒåé¡æã®ååšã®ã°ã©ãã¯æ¬¡ã®ããã«è¡šãããïŒ\r\n$$\\tag{1} (x-a)^2+y^2=(2-\\sqrt{a^2+1})^2$$\r\nã$0 \\leq a \\leq \\sqrt{3}$ ã®ç¯å²ã§ $a$ ãå¢å ããããšïŒåã®äžå¿ã¯å³ã«ç§»åããŠããïŒåã®ååŸã¯å調ã«æžå°ããïŒãã®ããšãã $x\\lt 0$ ã®ç¯å²ã§ã¯é å $x^2+y^2 \\leq 1$ ãééããŠããïŒ\\\r\nã次㫠$x \\geq 0$ ã®ç¯å²ãèããïŒ$x$ ãåºå®ããããšãã® $y$ ã®å€åãæ±ãããïŒæ¬åã§ããã° $y$ ã®æ倧å€ããæ±ããã°ååã§ããïŒ\\\r\nãåŒ $(1)$ ã $y^2=$ ã®åœ¢ã«å€åœ¢ãïŒ$a$ ã«ã€ããŠã®é¢æ°ã ãšèŠãŠæ¥µå€ãæ±ããïŒ\\\r\nã$\\dfrac{d}{da} y^2=0$ ã解ããš $a^2=\\dfrac{x^2}{4-x^2}$ ãåŸãïŒãããåŒ $(1)$ ã«ä»£å
¥ããŠå€åœ¢ãããšïŒæ¬¡ãåŸãïŒ$y=\\sqrt{4-x^2}-1$\\\r\nããã®åŒã¯ $x^2+(y-1)^2=4$ ãšæžãæããããšãå¯èœã§ããïŒãã£ãŠ $x$ ãåºå®ãããšãã® $y$ ã®æ倧å€ã®åè£ã¯ïŒä»¥äžã®ããããã§ããïŒ\r\n- $a=0, \\sqrt{3}$ ã«ãããåŒ $(1)$ ããæ±ãŸã $y$\r\n- åŒ $x^2+(y+1)^2=4$ ããæ±ãŸã $y$\r\n\r\nãå³ç€ºããã°ïŒåŸè
ã§ããããšã¯å®¹æã«ç¢ºãããããïŒ\\\r\nãããšã¯å¯Ÿç§°æ§ããæ±ããã¹ãé åããããã®ã§ïŒé¢ç©ãæ±ããã°ããïŒ\r\n\r\n---\r\n\r\nããªãæ¬è§£èª¬ã§ã¯éäžèšç®ãå²æãããïŒ$\\dfrac{d}{da} y^2=0$ ã解ãïŒãã®çµæãåŒ $(1)$ ã«ä»£å
¥ãããšããã¯ïŒãããªãã«èšç®ãé¢åã§ããïŒ",
"text": "代æ°çã«è§£ãæ¹æ³",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/4575/532"
},
{
"content": "æ¬è³ªçã«ã¯ [代æ°çã«è§£ãæ¹æ³](https:\\/\\/onlinemathcontest.com\\/contests\\/omcb012\\/editorial\\/4575\\/532) ãšå
šãçããã, å€æ°å€æãå³åœ¢çæŽå¯ãçšããã«, 埮åãå«ãåŒå€åœ¢ãšå Žååãã§å³å¯ã«è§£ãã.\r\n\r\n---\r\n\r\nååšã®æ¹çšåŒã¯\r\n$$\r\ny^2 = - \\left(x-\\tan{\\alpha}\\right)^2 + \\left(2-\\frac{1}{\\cos{\\alpha}}\\right)^2 \r\n$$\r\nã§è¡šãã, ãã®å³èŸºã $f(\\alpha, x)$ ãšãã. å $x$ ã«å¯ŸããŠ,\r\n$$\r\ng(x)=\\max_{\\alpha \\in \\left[0, \\ \\frac\\pi 3 \\right]}f(\\alpha, x)\r\n$$ ãç¥ããã.\r\n\r\n$$\r\n\\frac{\\partial f(\\alpha, x)}{\\partial \\alpha}=\\frac{2}{\\cos^2 \\alpha}\\left(x-2\\sin{\\alpha}\\right)\r\n$$\r\n\r\n$x\\lt 0$ ã®ãšã, $g(x)=f(0, x)=-x^2+1$.\\\r\n$0\\leq x\\leq\\sqrt{3}$ ã®ãšã, $g(x)=f\\left(\\arcsin{\\frac x 2}, x\\right)=\\left(\\sqrt{-x^2+4}-1 \\right)^2 \\geq 0$.\\\r\n$x\\gt \\sqrt{3}$ ã®ãšã, $g(x)=f\\left(\\frac\\pi 3, x\\right)=-\\left(x-\\sqrt 3 \\right)^2 \\lt 0$.\r\n\r\nååšã®ééé åã¯, $g(x)\\geq 0$ ãªã $x$ äžã§ $|y|\\leq \\sqrt{g(x)}$ ã§è¡šãããéšåã§, ãã®é¢ç©ã¯\r\n$$\r\n2\\left(\\int_{-1}^{0} \\sqrt{-x^2+1} \\ \\mathrm{d}x + \\int_{0}^{\\sqrt{3}} \\left(\\sqrt{-x^2+4}-1\\right) \\mathrm{d}x\\right)=\\frac{11}{6}\\pi-\\sqrt{3}\r\n$$\r\nã§ãã.",
"text": "é åæ³",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/4575/551"
}
] | ã$0\leq \alpha\leq \dfrac{\pi}{3}$ ã®ç¯å²ã§å®æ° $\alpha$ ãåãããšãïŒ $(\tan \alpha,0)$ ãäžå¿ã«æã€ååŸ $2-\dfrac{1}{\cos \alpha}$ ã®ååšãééããé åã®é¢ç©ãæ±ããŠãã ããïŒ\
ããã ãïŒæ±ããçãã¯ïŒæ£æŽæ° $a,b,c$ ãçšããŠïŒ$\dfrac{a}{b}\pi-\sqrt{c}$ ãšè¡šããã®ã§ïŒãã ã $a,b$ ã¯äºãã«çŽ ïŒïŒ$a+b+c$ ã®å€ã解çããŠãã ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/10741 | A | OMC221(A) | 100 | 337 | 364 | [
{
"content": "ãããããã®ã¢ã«ãã¡ãããã§æžãæããã®ã¡ã®æ°åãè¡šãããšãšããïŒ$a$ ã $3$ ã€ïŒ$r$ ã $2$ ã€ãããã®ä»ã®ã¢ã«ãã¡ããã㯠$1$ ã€ãã€ããããšããïŒ\r\n$$2a+r+(0+1+\\dots+9)=54$$ \r\nããªãã¡ $2a+r=9$ ã§ããïŒãŸãïŒ$a\\neq r$ ã§ããããïŒãããæºããã®ã¯\r\n$$(a,r)=(0,9),(1,7),(2,5),(4,1)$$\r\nã§ããïŒããããã«å¯ŸããŠæ®ã $8$ ã€ã®ã¢ã«ãã¡ãããã«ä»£å
¥ããæ¹æ³ã¯ $8!$ éãããã®ã§ïŒæ±ããå Žåã®æ°ã¯ $4\\times8!=\\mathbf{161280}$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/10741"
}
] | ãé»æ¿ã« $10$ çš®é¡ïŒ$13$ åã®ã¢ã«ãã¡ããã
$$m,a,r,t,h,s,a,k,u,r,a,n,o$$
ãæžãããŠããŸã ($a$ ã $3$ ã€ïŒ$r$ ã $2$ ã€ïŒä»ã®ã¢ã«ãã¡ããã㯠$1$ ã€ãã€æžãããŠããŸã) ïŒä»¥äžãã¿ããããã«ããããã®ã¢ã«ãã¡ãããã $0$ ä»¥äž $9$ 以äžã®æŽæ°ã«æžãæããæ¹æ³ã¯ããã€ãããŸããïŒ
- åãã¢ã«ãã¡ãããã¯åãæ°åã«ïŒç°ãªãã¢ã«ãã¡ãããã¯ç°ãªãæ°åã«æžãæããïŒ
- å
šãŠã®ã¢ã«ãã¡ããããæžãæããã®ã¡ïŒé»æ¿ã«æžãããæ°åã®å㯠$54$ ã§ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/7797 | B | OMC221(B) | 200 | 198 | 272 | [
{
"content": "**解æ³1.**ãå¹³è¡å蟺圢ã®èŸºã®é·ãã $x, y$ ãšãããšïŒäœåŒŠå®çãã\r\n$$x^2+y^2-2xy\\cos89^\\circ=199^2,\\quad x^2+y^2+2xy\\cos89^\\circ=201^2$$\r\nã§ããããïŒ\r\n$$xy=\\dfrac{201^2-199^2}{4\\cos89^\\circ}=\\dfrac{200}{\\cos89^\\circ}$$\r\nãåŸãïŒãã£ãŠïŒ\r\n$$S=xy\\sin89^\\circ=200\\tan89^\\circ$$\r\nãšè¡šããïŒãŸãïŒé è§ã $2^\\circ$ïŒåºèŸºã®é·ãã $1$ ã§ããäºç蟺äžè§åœ¢ã®é¢ç©ã¯ $\\tan89^\\circ\\/4$ ã§ããïŒ$T$ ã¯ããã $180$ åãããã®ã§ããããïŒ$T=45\\tan89^\\circ$ ãšè¡šããïŒãããã£ãŠïŒ$\\dfrac{S}{T}=\\dfrac{40}{9}$ ã§ããïŒç¹ã«è§£çãã¹ãå€ã¯ $\\mathbf{49}$ ã§ããïŒ\r\n\r\n**解æ³2.**ãå¹³è¡å蟺圢ã察è§ç·ã§äºçåããŠåŸããã $2$ çš®é¡ã®äžè§åœ¢ïŒå³ã® $P, Q$ïŒã $180$ åãã€çšæããŠïŒå³ã®ããã«äºãéãã«äžŠã¹ããšïŒäžèŸºã®é·ãã $201$ ã®æ£ $180$ è§åœ¢ããäžèŸºã®é·ãã $199$ ã®æ£ $180$ è§åœ¢ãããæããå³åœ¢ãåŸãïŒãã£ãŠïŒ\r\n$$S\\times180=T\\times(201^2-199^2)$$\r\nãæãç«ã¡ïŒ$\\dfrac{S}{T}=\\dfrac{40}{9}$ ãšãããïŒ\r\n",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/7797"
}
] | ãå
è§ã®ã²ãšã€ã $89^\circ$ ã§ãã£ãŠïŒ$2$ æ¬ã®å¯Ÿè§ç·ã®é·ãããããã $199$ ãš $201$ ã§ããå¹³è¡å蟺圢ã®é¢ç©ã $S$ïŒäžèŸºã®é·ãã $1$ ã§ããæ£ $180$ è§åœ¢ã®é¢ç©ã $T$ ãšãããšãïŒ$\dfrac{S}{T}$ ãæ±ããŠãã ããïŒãã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/9871 | C | OMC221(C) | 400 | 115 | 191 | [
{
"content": "ã$X_n=\\\\{1,1,2,2,\\dots,n,n\\\\}$ ãšããïŒ$f$ ã®å®çŸ©ããã®éšåéåã«æ¡åŒµããïŒããã§ïŒ$f(\\emptyset)=1$ ãšããïŒãã®ãšãïŒ$n=2,3,\\dots$ ã«ã€ããŠïŒ$X_{n-1}$ ã®éšåéå $V$ ãš $\\\\{n,n\\\\}$ ã®éšåéåã®åéåã«ã€ããŠïŒä»¥äžãæç«ããïŒ\r\n$$\r\n\\begin{aligned}\r\nf(V\\cup \\emptyset)&=f(V),\\\\\\\\\r\nf(V\\cup \\\\{n\\\\})&=\r\n\\begin{cases}\r\nf(V)&(Vã®èŠçŽ æ°ãå¥æ°)\\\\\\\\\r\nnf(V)&(Vã®èŠçŽ æ°ãå¶æ°)\r\n\\end{cases},\\\\\\\\\r\nf(V\\cup \\\\{n,n\\\\})&=nf(V).\r\n\\end{aligned}\r\n$$\r\nããã£ãŠïŒ$X_n$ ã®éšåéå $V$ ã®ãã¡ïŒèŠçŽ æ°ãå¶æ°ã§ãããã®ã® $f(V)$ ã®ç·åã $a_n$ïŒèŠçŽ æ°ãå¥æ°ã§ãããã®ã® $f(V)$ ã®ç·åã $b_n$ ãšãããšïŒ$a_1=2, ~ b_1=1$ ããã³ $n=2,3,\\dots$ ã«ã€ããŠä»¥äžãæç«ããïŒ\r\n$$\r\na_n=(1+n)a_{n-1}+b_{n-1}, \\quad b_n=na_{n-1}+(1+n)b_{n-1}.\r\n$$\r\nãã£ãŠïŒæ±ãããã®ã¯ $a_{5}+b_{5}-1=1729+3430-1=\\mathbf{5158}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/9871"
}
] | ãå€ééå $\\{1,1,2,2,3,3,4,4,5,5\\}$ ã®ç©ºã§ãªãéšåéå $U$ ã«ã€ããŠïŒãã®èŠçŽ ãæé ã«äžŠã¹ããšãå¥æ°çªç®ã«ããããã®ã®ç·ç©ã $f(U)$ ãšããŸãïŒäŸãã° $U=\\{1,2,2,4,5,5\\}$ ã®ãšãïŒ$f(U)=1\times2\times5$ ã§ãïŒ$U$ ãšããŠèãããããã®ã¯ $3^{5}-1$ éããããŸããïŒããã§ïŒåãæ°ã¯åºå¥ããªããã®ãšããŸãïŒïŒããããã¹ãŠã«å¯Ÿãã $f(U)$ ã®ç·åãæ±ããŠãã ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/5046 | D | OMC221(D) | 400 | 10 | 47 | [
{
"content": "ã$\\angle DAP+\\angle PRD=180^\\circ$ ãã $A, P, R, D$ ã¯å
±åã§ããïŒååšè§ã®å®çãã $\\angle RAP=\\angle RDP=\\angle RPD=\\angle RAD$ ã§ããããïŒ$R$ 㯠$AC$ äžã«ããïŒ\\\r\nã$PD$ ãš $AC$ ã®äº€ç¹ã $S$ ãšããïŒ$\\triangle BRS$ ã $AC$ ã«é¢ããŠç·å¯Ÿç§°ç§»åããå³åœ¢ã $\\triangle DRS$ ã§ããïŒãã£ãŠïŒ$\\angle RBS=\\angle RDS=\\angle RPS$ ãšãªããã $B, R, S, P$ ã¯å
±åã§ããïŒ\\\r\nã$AP=x\\ (0\\leq x \\leq 1)$ ãšããïŒ$AB \\parallel CD$ ããïŒ\r\n$$AS:SC=AP:CD=x:1,\\quad AR:RC=AB:CQ=1:\\bigg(1-\\dfrac{9}{10}x\\bigg)$$\r\nãã£ãŠïŒ$AC=k(x+1)\\bigg(2-\\dfrac{9}{10}x\\bigg)\\ (k\\geq 0)$ ãšããã°ïŒ$AS=kx \\bigg(2-\\dfrac{9}{10}x \\bigg),\\ AR=k(x+1)$ïŒæ¹ã¹ãã®å®çãã $AP \\times AB=AS \\times AR$ ã§ããããïŒ\r\n$$k=\\cfrac{1}{\\sqrt{(x+1) \\bigg(2-\\cfrac{9}{10}x \\bigg)}}$$\r\nåŸã£ãŠïŒ\r\n$$AC=\\sqrt{(x+1) \\bigg(2-\\dfrac{9}{10}x \\bigg)}$$\r\nã§ããïŒãã®å€ã¯ $x=\\dfrac{11}{18}$ ã®ãšãïŒæå€§å€ $\\dfrac{29 \\sqrt{10}}{60}$ ããšãïŒãã£ãŠïŒç¹ã«è§£çãã¹ãå€ã¯ $\\bm {99}$ïŒ\r\n\r\n",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/5046"
},
{
"content": "ãå
¬åŒè§£èª¬åæ§, $A,R,C$ ã¯å
±ç·ã§ãã, $AC$ 㯠$BD$ ã®åçŽäºçåç·ã§ãããã, $RD=RB$ ãæãç«ã€.\r\nãããã£ãŠ, $AP=10x(\\iff DQ=9x)$ ãšã, $R$ ãã $AB,CD$ ã«äžãããåç·ã®è¶³ããããã $H_1,H_2$ ãšãããš, $PR=RB$ ãã, \r\n$$PH_1=H_1B=\\frac{1}{2}PB=\\frac{1}{2}-5x$$\r\n$\\triangle RAB$ ãš $\\triangle RCQ$ ã¯çžäŒŒã§, $H_1,H_2$ ã¯ãããã®çžäŒŒã«ãããŠ, ãããã察å¿é¢ä¿ã«ãããã,\r\n$$H_2Q=\\frac{CQ}{AB}H_1B=\\Big(1-9x\\Big)\\Big(\\frac{1}{2}-5x\\Big)$$\r\nãããã£ãŠ, \r\n$$\r\n\\begin{aligned}\r\nAH_1&=AP+PH_1=5x+\\frac{1}{2}\\\\\\\\\r\nDH_2&=DQ+QH_2=45x^2-\\frac{1}{2}x+\\frac{1}{2}\r\n\\end{aligned}\r\n$$\r\nãæç«ã, $\\angle DAB=\\theta$ ãšãããš, $AH_1=DH_2+AD\\cos{\\theta}$ ãæç«ã, ãããæŽçãããš, $\\cos{\\theta}$ ãš $x$ ã«ã€ããŠ, 以äžãåŸã.\r\n$$\\cos{\\theta}=-45x^2+\\frac{11}{2}x$$\r\n$0\\lt 10x \\lt 1$ ã®ç¯å²ã§äžã®åŒã¯ $x=\\dfrac{11}{180}$ ã®ãšã, æå€§å€ $\\dfrac{121}{720}$ ããšã, å®é $x=\\dfrac{11}{180}$ ãæºããå³ã¯ååšããã®ã§, \r\n$$AC=2AB\\cos{\\dfrac{\\theta}{2}}=2\\sqrt{\\frac{\\cos{\\theta}+1}{2}}$$\r\nã¯, $\\cos{\\theta}=\\dfrac{121}{720}$ ã®ãšãæå€§å€ $\\dfrac{29\\sqrt{10}}{60}$ ããšã.",
"text": "å¥è§£",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/5046/520"
},
{
"content": "ãã©ã¹ãã®äºæ¬¡åŒç«ãŠã®ãšãã«Ptolemyã®å®çãçšããæ¹æ³ã§ãïŒ\r\n\r\n---\r\n\r\nã$A, P, R, D$ ãå
±åã§ããããšïŒããã³ $R$ ã $AC$ äžã«ããããšã¯å
¬åŒè§£èª¬ãšåæ§ã«ç¢ºãããããïŒãŸãïŒ$AB \\parallel DC$ ãã $\\triangle ABR \\sim \\triangle CQR$ ãå°ãããšãã§ãïŒ$AB = BC, PR = RD, \\angle ABC = \\angle PRD$ ãã $\\triangle ABC \\sim \\triangle PRD$ ãå°ãããšãã§ããïŒ \\\r\nã$CQ = x, AC = y$ ãšããïŒãããš\r\n$$AP = \\frac{10}{9}DQ = \\frac{10}{9}(1 - x)$$\r\nãåŸããïŒ$\\triangle ABR, \\triangle CQR$ ã®çžäŒŒæ¯ã $1 : x$ ã§ããããšãã\r\n$$AR = \\frac{1}{x + 1}AC = \\frac{y}{x + 1}$$\r\nãåŸããïŒããã« $\\triangle ABC \\sim \\triangle PRD$ ãã\r\n$$\\frac{DP}{PR} = \\frac{AC}{AB} = y$$\r\nãåŸãããïŒåè§åœ¢ $APRD$ ã«é¢ããŠPtolemyã®å®çãçšãããš\r\n$$AR \\times DP = AD \\times PR + AP \\times RD = (AD + AP) \\times PR$$\r\nããïŒ\r\n$$\\frac{y^2}{x + 1} = AR \\times \\frac{DP}{PR} = AD + AP = \\frac{10}{9}(1 - x) + 1$$\r\nãæãç«ã€ã®ã§ïŒãããã\r\n$$y = \\frac{1}{3}\\sqrt{(19 - 10x)(x + 1)}$$\r\nãåŸãïŒãã®å³èŸºã¯ $x = \\dfrac{9}{20}$ ã§æå€§å€ $\\dfrac{29\\sqrt{10}}{60}$ ããšãïŒ$x$ ããã®å€ã®ãšãã®å³ã¯ç¢ºãã«ååšããïŒïŒãã£ãŠïŒè§£çãã¹ãå€ã¯ $\\mathbf{99}$ ã§ããïŒ",
"text": "Ptolemyã®å®çã䜿ã",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/5046/528"
},
{
"content": "ã$A,R,C$ ãå
±ç·ã§ããããšã®å¥èšŒæã玹ä»ããïŒ\\\r\nãç¹ $P$ ã蟺 $AB$ äžãåãïŒããã«åãããŠç¹ $R$ ãæ¡ä»¶ãæºããããã«åãããšãèããïŒ\\\r\nããã®ãšã $\\overrightarrow{DR}$ ã¯ïŒ$\\overrightarrow{DP}$ ãå®æ°åããŠïŒç¹ $D$ äžå¿ã«äžå®ã®è§åºŠå転ããããã¯ãã«ã§ããïŒåã®è¯ã人ã§ããã°ïŒä»¥äžã®èª¬æã ãã§ïŒç¹ $R$ ãããçŽç·äžã«ååšãããšçŽèŠ³ãããããããªãïŒã¡ãªã¿ã«ç§ã¯åãè¯ããªãïŒïŒ\\\r\nãããå
·äœçã«èª¬æãããšïŒâ ç¹ $D$ ãåç¹ãšããçŽäº€åº§æšç³»ãåãã°ïŒç¹ $P,R$ ã¯ããè¡åã«ãã£ãŠåº§æšå€æããé¢ä¿ã ãšèšãããïŒâ¡ç¹ $D$ ãåç¹ãšããè€çŽ æ°å¹³é¢ãåãã°ïŒç¹ $P,R$ ã¯ããè€çŽ æ°åããé¢ä¿ã ãšèšããïŒãããã«ããŠãïŒçŽç·ãçŽç·ã«ç§»ãå€æã§ããããšãèšããïŒ\\\r\nãããšã¯ïŒç¹ $P$ã ç¹ $A$ ã«äžèŽããå ŽåïŒç¹ $P$ ãç¹ $B$ ã«äžèŽããå Žåã«ã€ããŠç¢ºãããã°ïŒ$A,R,C$ ãå
±ç·ã§ããããšã瀺ããïŒ\r\n\r\n---\r\n\r\nããã£ãããªã®ã§ïŒãã®åŸã®å¥è§£ã«ã€ããŠïŒãã¯ãã«ã䜿ã£ããã®ãæžããŠããïŒ\\\r\nã$\\overrightarrow{AB}=\\overrightarrow{b},\\overrightarrow{AD}=\\overrightarrow{d}$ ãšããïŒå®æ° $s,t$ ãçšããŠã$\\overrightarrow{AP}=10t\\overrightarrow{b},\\overrightarrow{AQ}=9t\\overrightarrow{b}+\\overrightarrow{d},\\overrightarrow{AR}=s(\\overrightarrow{b}+\\overrightarrow{d})$ ãšãããïŒ$0 \\lt s \\lt 1$ïŒ$0 \\lt t \\lt \\dfrac{1}{10}$ã§ããïŒïŒ\\\r\nã$\\overrightarrow{BR} \\ \\/\\/ \\ \\overrightarrow{BQ}$ ãçšããã°æ¬¡ã®åŒãåŸãïŒ\r\n$$t=\\dfrac{2s-1}{9s}$$\r\nããŸã $|\\overrightarrow{DR}|^2=|\\overrightarrow{DP}|^2$ ãçšããã°æ¬¡ã®åŒãåŸãïŒ$\\overrightarrow{b} \\cdot \\overrightarrow{d}=\\alpha$ ãšãããïŒïŒ\r\n$$\\alpha=\\dfrac{10t-2s+1}{2s}$$\r\nã以äžã® 2 åŒãçšããã°ïŒæ¬¡ã®ããã«èšç®ã§ããïŒ\r\n$$\\alpha=-\\dfrac{5}{9} \\left( \\dfrac{1}{s}- \\dfrac{29}{20}\\right)^2+\\dfrac{121}{720}$$\r\nã$AC^2=2+2\\alpha$ ãã $\\alpha=\\dfrac{121}{720}$ ã®ãšã $AC$ ã¯æ倧ã§ããïŒ",
"text": "A,R,C ãå
±ç·ã§ããããšã®å¥èšŒæãªã©",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/5046/529"
}
] | ãäžèŸºã®é·ãã $1$ ã®ã²ã圢 $ABCD$ ãããïŒèŸº $AB$ äžã«ç¹ $P$ïŒèŸº $CD$ äžã«ç¹ $Q$ïŒç·å $BQ$ äžã«ç¹ $R$ ããšããŸãïŒããŸïŒ
$$AP:DQ=10:9,\quad PR=RD,\quad \angle ABC=\angle PRD$$
ãæãç«ã€ãšãïŒç·å $AC$ ã®é·ããšããŠããããæ倧å€ãæ±ããŠãã ããïŒååšãä¿èšŒãããŸãïŒïŒãã ãïŒæ±ããå€ã¯æ£æŽæ° $a, b, c$ïŒ$a, c$ ã¯äºãã«çŽ ã§ããïŒ$b$ ã¯å¹³æ¹å åããããªãïŒãçšã㊠$\dfrac{a \sqrt b}{c}$ ãšè¡šãããã®ã§ïŒ$a+b+c$ ã解çããŠãã ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/5047 | E | OMC221(E) | 400 | 58 | 110 | [
{
"content": "ã$X^2$ ã®æé«äœã®æ°ãåãé€ããŠã§ããæ°ãïŒæŽæ° $Y$ ãçšã $Y^2$ ãšããïŒãŸãæ¡ä»¶ãé ã« (a),(b),(c) ãšããïŒ$X\\geq4$ ãš (b),(c) ããïŒããæ£æŽæ° $n$ ãååšããŠ\r\n$$(X+Y)(X-Y)=2^{n+3}\\times5^n\\quad\\cdots(1),\\qquad10^n\\gt Y^2\\quad\\cdots(2)$$\r\nãæãç«ã€ïŒãã®ãšã $n+1$ 㯠$X^2$ ã®æ¡æ°ãšäžèŽããïŒïŒ\r\nç°¡åãªè°è«ã«ããïŒ(1) ãã¿ãã $(X,Y)$ ã®çµã¯ãã $i,j\\ (0\\leq i\\leq n+1,0\\leq j\\leq n)$ ãçšããŠæ¬¡ã®åœ¢ã«è¡šããããšãããïŒ\r\n$$(X,Y)=(2^i\\times 5^j+2^{n+1-i}\\times 5^{n-j},2^i\\times 5^j-2^{n+1-i}\\times 5^{n-j})$$\r\n\r\nããã (2) ã«ä»£å
¥ããŠæŽçããã°æ¬¡ãåŸããã ($t = 2^{2i}5^{2j}$ ãšããïŒ$t$ ã®äºæ¬¡äžçåŒã解ãããšã§åŸããã) ïŒ\r\n$$10^n\\lt 2^{2i}\\times 5^{2j}\\lt 4\\times 10^n\\quad\\cdots(3)$$\r\n\r\nãŸã (a) ãã $X$ 㯠$10$ ã®åæ°ã§ãªãããïŒ$i=0,i=n+1,j=0,j=n$ ã®ãã¡å°ãªããšãäžã€ãæç«ããªããã°ãªããªãïŒå¿
èŠãªãã° $(i,j)$ ã $(n+1-i,n-j)$ ã§çœ®ãæããããšã§ $i=0$ ããã³ $j=0$ ã®å Žåã®ã¿èããã°ããïŒãŸã $j=0$ ã®å Žå (3) ãã$10^n\\lt 2^{2i}\\lt 4\\times 10^n$ ã§ãã $i\\leq n+1$ ã«æ³šæããã°ãããã¿ããã®ã¯ $(n,i)=(1,2)$ ã®ã¿ã ãïŒãã㯠$i=n+1$ ãã¿ãããŠãããã $i=0$ ã®å Žåã«åž°çãããïŒãã£ãŠçµå± $i=0$ ãšããŠæ§ããªãïŒ\\\r\nã(3) ãã$10^n\\lt 5^{2j}\\lt 4\\times 10^n$ ã§ããïŒäž¡èŸºã®åžžçšå¯Ÿæ°ãåã $2\\log_{10}2\\lt 1$ ã«æ³šæããã°ïŒ\r\n$$0\\lt\\\\{2(1-\\log_{10}2)j\\\\}\\lt 2\\log_{10}2\\quad\\cdots(4)$$ ãã¿ãã $j$ ãèããã°ããïŒãã ã $\\\\{x\\\\}$ ã§å®æ° $x$ ã®å°æ°éšåãè¡šãïŒãã£ãŠïŒ$j$ ãå°ããæ¹ããè©Šãããšã§ïŒ$6$ çªç®ã«å°ãã(4)ãæºãã $j$ 㯠$9$ ã§ããããšãåãããïŒããäžè¬ã«ïŒ$j\\leq 49$ ã®ç¯å²ã§(4)ãæºãã $j$ 㯠$j\\equiv 1,3,4 \\pmod{5}$ ã§ããããšã以äžã®ããã«ããŠç€ºãããšãã§ããïŒ\\\r\nã$j\\le 49$ ã§ã¯æ¬¡ãæãç«ã€ïŒ\r\n\r\n$$1.4j-0.196\\leq 1.396j\\lt 2(1-\\log_{10}2)j\\lt 1.398j\\leq 1.4j$$\r\n\r\n$\\\\{1.4j\\\\}$ ã®ãšãåŸãå€ã¯ $0,0.2,0.4,0.6,0.8$ ã§ããããšã«æ³šæããã°ïŒ$j\\leq 49$ ãªãã°æ¬¡ãæãç«ã€ïŒ\r\n$$\\\\{1.4j-0.196\\\\}\\lt\\\\{2(1-\\log_{10}2)j\\\\}\\lt\\\\{1.4j-0.196\\\\}+0.196$$\r\n\r\n$0.602\\lt 2\\log_{10}2\\lt 0.604$ ã«æ³šæããã°ïŒ$j\\leq 49$ ã®ç¯å²ã§ (4) ãã¿ãã $j$ 㯠$j\\bmod{5}=1,3,4$ ãªã $j$ ã§ããããšããããïŒ\\\r\nã以äžããïŒæ±ããå€ã¯ $j = 9$ ã®ãšãã«åŸããïŒãã®ãšã $n = 12$ ã§ããããïŒæ±ããå€ã¯ $X=5^9+2^{13}\\times 5^3={\\bf 2977125}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/5047"
},
{
"content": "ã$(X, Y) = (5^j + 2^{n+1} \\times 5^{n-j}, 5^j - 2^{n+1} \\times 5^{n-j})$ ãŸã§ã¯çç¥(å
¬åŒè§£èª¬ãåç
§, $j=0 \\Rightarrow (n, i) = (1, 2)$ ã«é¢ããŠã¯äžçåŒ $(3)$ ã䜿ããªããŠã $Y^2 \\lt 10^n$ ããåãã)ïŒ$Y^2 \\lt 10^n$ ã« $Y = 5^j - 2^{n+1} \\times 5^{n-j}$ ã代å
¥ãããš\r\n$$5^{2j} - 2^{n+2} \\times 5^n + 2^{2n+2} \\times 5^{2n-2j} \\lt 10^n\\\\\\\\\r\n\\Leftrightarrow 5^{2j - 1} + 2^{2n+2} \\times 5^{2n-2j - 1} \\lt 10^n\\\\\\\\\r\n\\Rightarrow 5^{2j - 1} \\lt 10^n, \\quad 2^{2n+2} \\times 5^{2n-2j-1} \\lt 10^n$$\r\nãåŸã $2$ åŒã«ã€ããŠïŒãããã䞡蟺ã®å¯Ÿæ°ããšããšïŒ\r\n$$(2j-1)(1- \\log _{10} 2) \\lt n, \\quad (2n+2) \\log _{10} 2 + (2n-2j-1)(1 - \\log _{10} 2) \\lt n\\\\\\\\\r\n\\Leftrightarrow (2j - 1)(1 - \\log _{10} 2) \\lt n \\lt 2j+1 - (2j+3) \\log _{10} 2\\\\\\\\\r\n\\Rightarrow 0.698(2j-1) \\lt n \\lt 2j+1 - 0.301(2j+3)$$\r\nããããæºãã $(j, n)$ ã®çµãæé ã«äžŠã¹ããš\r\n$$(j, n) = (1, 1), (3, 4), (4, 5), (5, 7), (6, 8), (8, 11), (9, 12), \\ldots$$\r\nã$Y^2 \\lt 10^n$ ãæºãããé ã«èª¿ã¹ãããšã§ïŒ$(j, n) = (5, 7)$ ãäžé©ã§ $(j, n) = (9, 12)$ ã $6$ çªç®ã«å°ãã $X$ ãäžããçµã ãšåããïŒ",
"text": "å
¬åŒè§£èª¬ã®äžçåŒ(3)ãå°ããªãã£ãå Žå",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/5047/526"
}
] | ã次ã®æ¡ä»¶ããã¹ãŠæºãã $4$ 以äžã®æŽæ° $X$ ãšããŠããåŸãå€ã®ãã¡ $6$ çªç®ã«å°ãããã®ãæ±ããŠãã ããïŒãã ãïŒæ¡ä»¶ã¯ãã¹ãŠåé²æ³è¡šèšã§èããŸãïŒ
- $X^2$ ã® $1$ ã®äœã¯ $0$ ã§ãªãïŒ
- $X^2$ ã®æé«äœã®æ°ã¯ $8$ ã§ããïŒ
- $X^2$ ã®æé«äœã®æ°ãåãé€ããŠã§ããæ°ã¯å¹³æ¹æ°ã§ããïŒ
ãäŸãã°ïŒ$10201$ ã®æé«äœã®æ°ãåãé€ããŠã§ããæ°ã¯ $201$ ã§ãïŒãŸãïŒ$0.301 \lt\log_{10}2 \lt 0.302$ ãæãç«ã¡ãŸãïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/10100 | F | OMC221(F) | 600 | 11 | 34 | [
{
"content": "ã$P$ ã®ç§»åã«ä»¥äžã®ããã«ååãã€ããïŒ\r\n- æäœ $X$ : ç¹ $P$ ã $x$ æ¹åã« $1$ ã ã移åããã\r\n- æäœ $Y$ : ç¹ $P$ ã $y$ æ¹åã« $1$ ã ã移åããã\r\n- æäœ $Z$ : ç¹ $P$ ã $z$ æ¹åã« $1$ ã ã移åããã\r\n\r\nã$P$ ã®ç§»åæ¹æ³ã«å¯Ÿã㊠$X,Y,Z$ ãããªãæååã察å¿ãããããšãèããïŒããšãã°ïŒæäœ $X$ $\\rightarrow$ æäœ $Y$ $\\rightarrow$ æäœ $Z$ $\\rightarrow$ $\\cdots$ã«å¯ŸããŠã¯ïŒ$XYZ\\cdots$ ãšå¯Ÿå¿ãããïŒããŸïŒ$X,Y$ ãããªãæååã«å¯ŸãïŒ$X,Y$ ãããããäžåãã€éžãã§ïŒ$Z$ ã«çœ®ãæããããšãæäœ $Q$ ãšåŒã¶ããšã«ããïŒä»¥äžã®è£é¡ãæãç«ã€ïŒ\r\n\r\n---\r\n**è£é¡ïŒ**\r\n$n$ åã® $X$ ãš $m$ åã® $Y$ ã䞊ã³æ¿ããŠã§ãããã¹ãŠã®æååã«ã€ããŠïŒæäœ $Q$ ãè¡ããããã¹ãŠã®å Žæã« $1$ 床ãã€è¡ã£ããšãïŒ$n-1$ åã® $X$ ãš $m-1$ åã® $Y$ ïŒ$2$ åã® $Z$ ã䞊ã³æ¿ããŠã§ããæååãã¹ãŠããããã $2$ åãã€ç»å ŽããïŒ\r\n\r\n**蚌æïŒ**\r\n$n-1$ åã® $X$ ãš $m-1$ åã® $Y$ ïŒ$2$ åã® $Z$ ã䞊ã³æ¿ããŠã§ããæåå $T$ ã«å¯ŸãïŒ$T$ ã® $X,Y$ ãããäœçœ®ã«ã¯ãããã $X,Y$ ãäœçœ®ãïŒ$2$ ã€ã® $Z$ ã®äœçœ®ã«ã¯ïŒæåããé ã« $X,Y$ ãŸã㯠$Y,X$ ãäœçœ®ãããã㪠$2$ ã€ã®æååã察å¿ããããïŒè£é¡ãåŸãïŒ\r\n\r\n---\r\n\r\nãããã§æ±ããã¹ã³ã¢ã®ç·åã¯æäœæ¹æ³ã«å¯Ÿå¿ãããæååãã¹ãŠã«ã€ããŠïŒããã«å«ãŸãã $100$ åã® $Z$ ã§åºåã£ãŠã§ãã $101$ åã® $X,Y$ ãããªãéšåæååã«å¯ŸãïŒæäœ $Q$ ãè¡ãæ¹æ³ã®ç·æ°ã§ããïŒäžã®äºå®ããïŒãã®ãšã $400-101$ åã®$X$ ãšïŒ$400-101$ åã® $Y$ ïŒ$100+101+101$ åã® $Z$ ã䞊ã³æ¿ããŠã§ããæååãã¹ãŠã $2^{101}$ åãã€ç»å Žãã (æäœ $Q$ ããã®åŸ©å
ã®éïŒ$3k$ æåç®ã® $Z$ 㯠$X, Y$ ã«å€æŽãããªãããšã«æ³šæ)ïŒãã£ãŠïŒ$$M=\\dfrac{900!}{299!299!302!}\\times2^{101}$$ ã§ããïŒããã $2,3$ ã§å²ãåããæ倧ã®åæ°ã¯ãããã $112,6$ ã§ããïŒãã£ãŠïŒ$\\gcd(M,6^M)=2^{112}3^6$ ã§ããïŒè§£çãã¹ãå€ã¯ $(112+1)(6+1)=\\mathbf{791}$ ãšãªãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/10100"
},
{
"content": "ã$k=0,1,\\dots,100$ ã«ã€ããŠ å¹³é¢ $z=k$ å
ã§, $x,y$ æ¹åã« $1$ é²ãåæ°ããããã $x_k,y_k$ ãšãããš, \r\nã¹ã³ã¢ã¯ $\\prod_{k=0}^{100}x_ky_k$ ãšãªã, ãã®ãããªæäœæ¹æ³ã¯ $\\prod_{k=0}^{100}\\binom{x_k+y_k}{x_k}$ éããšãªã.\r\nãããã£ãŠ, \r\n$$\r\n\\begin{aligned}\r\nx_0+x_1+\\dots+x_{100}=400,\\\\\\\\\r\ny_0+y_1+\\dots+y_{100}=400\r\n\\end{aligned}\r\n$$\r\nãæºããéè² æŽæ°ã®çµ $(x_0,x_1,\\dots,x_{100},y_0,y_1,\\dots,y_{100})$ ãã¹ãŠã«ã€ããŠ, \r\n$$\\Bigg(\\prod_{k=0}^{100}x_ky_k\\Bigg)\\Bigg(\\prod_{k=0}^{100}\\binom{x_k+y_k}{x_k}\\Bigg)=\\Bigg(\\prod_{k=0}^{100}x_ky_k\\binom{x_k+y_k}{x_k}\\Bigg)$$\r\nã®ç·åãæ±ããã°è¯ã.\\\r\nã$x,y$ ã® $2$ å€æ°åªçŽæ°ã«ã€ããŠä»¥äžãæç«ãã. \r\n$$f(x,y)=\\dfrac{1}{1-(x+y)}=\\sum_{k=0}^{\\infty}(x+y)^k=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}\\binom{i+j}{i}x^iy^j$$\r\nãããçšããŠ, $g(x,y)=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}ij\\binom{i+j}{i}x^iy^j$ ã¯ä»¥äžã®ããã«è¡šç€ºã§ãã.\r\n$$\r\n\\begin{aligned}g(x,y)&=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}ij\\binom{i+j}{i}x^iy^j\\\\\\\\\r\n&=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}xy\\frac{\\partial}{\\partial y}\\frac{\\partial}{\\partial x}\\Bigg(\\binom{i+j}{i}x^iy^j\\Bigg)\\\\\\\\\r\n&=xy\\frac{\\partial}{\\partial y}\\frac{\\partial}{\\partial x}\\Bigg(\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}\\Bigg(\\binom{i+j}{i}x^iy^j\\Bigg)\\Bigg)\\\\\\\\\r\n&=xy\\frac{\\partial}{\\partial y}\\frac{\\partial}{\\partial x}\\Bigg(\\frac{1}{1-x-y}\\Bigg)\\\\\\\\\r\n&=\\frac{2xy}{(1-x-y)^3}\r\n\\end{aligned}$$\r\nã$M$ 㯠$(g(x,y))^{101}$ ã® $x^{400}y^{400}$ ã®ä¿æ°ã§ãã, 以äžã®ããã«èšç®ã§ãã.\r\n$$\r\n\\begin{aligned}\r\n\\[x^{400}y^{400}\\](g(x,y))^{101}&=\\[x^{400}y^{400}\\]\\frac{(2xy)^{101}}{(1-x-y)^{303}}\\\\\\\\\r\n&=2^{101}\\[x^{400-101}y^{400-101}\\]\\frac{1}{(1-(x+y))^{303}}\\\\\\\\\r\n&=2^{101}\\[x^{299}y^{299}\\]\\Bigg(\\sum_{k=0}^{\\infty}\\binom{k+302}{302}(x+y)^k\\Bigg)\\\\\\\\\r\n&=2^{101}\\[x^{299}y^{299}\\]\\Bigg(\\binom{(299+299)+302}{302}(x+y)^{299+299}\\Bigg)\\\\\\\\\r\n&=2^{101}\\times\\binom{900}{302}\\binom{598}{299}\\\\\\\\\r\n&=2^{101}\\times\\frac{900!}{302!299!299!}\\\\\\\\\r\n\\end{aligned}\r\n$$\r\nãšãªã.",
"text": "å¥è§£",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/10100/521"
},
{
"content": "ãå¹³é¢ $z = k$ äžã§ $x$ 軞æ¹åã« $x_k$, $y$ 軞æ¹åã« $y_k$ ã ã移åãããšãããšïŒ$\\displaystyle\\sum_{k=0}^{100} x_k = 400, \\sum_{k=0}^{100} y_k = 400$ ããã³ $S_k = x_k y_k$ ãæç«ãïŒ$x_k, y_k \\gt 0$ ãšããŠããïŒ$\\lbrace x_k \\rbrace_{k = 0}^{100}$ ãš $\\lbrace y_k \\rbrace_{k = 0}^{100}$ ã®çµããããã«å¯Ÿå¿ãã移åæ¹æ³ã¯ $\\displaystyle\\prod_{k=0}^{100} \\dfrac{(x_k + y_k)!}{x_k! y_k!}$ éãååšããããïŒã¹ã³ã¢ãžã®å¯äžã¯\r\n$$\\prod_{k=0}^{100} \\dfrac{(x_k + y_k)!}{x_k! y_k!} \\prod_{k=0}^{100} S_k = 2^{101} \\times \\prod_{k=0}^{100} \\dfrac{(x_k + y_k)!}{(x_k - 1)! (y_k - 1)! 2!}$$\r\nãšãªãïŒäžåŒã®å³èŸºã®ç·ç©éšåã¯ïŒåç¹ $(0, 0, 0)$ ããç¹ $(299, 299, 302)$ ãžã®ç§»åã§ãã£ãŠïŒå $m = 1, 2, \\ldots, 100$ ã«ã€ããŠå¹³é¢ $z = 3m - 1$ ããå¹³é¢ $z = 3m$ ãžç§»åãããšãã® $x, y$ 座æšã $\\displaystyle\\sum_{k = 0}^{m - 1} (x_k - 1), \\displaystyle\\sum_{k = 0}^{m - 1} (y_k - 1)$ ãšãªããã®ã®ç·æ°ã«çããããïŒ$M = 2^{101} \\times \\dfrac{900!}{299!299!302!}$ïŒ",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc221/editorial/10100/523"
}
] | ã座æšç©ºéå
ã®ç¹ $P$ ãã¯ããåç¹ $(0,0,0)$ ã«ãããŸãïŒ$P$ ã $x, y, z$ ã®ããããã®æ£æ¹åã« $1$ ã ã移åãããæäœãèš $900$ åè¡ãããšã§ïŒç¹ $(400,400,100)$ ã«ç§»åãããããšãèããŸãïŒ\
ã$k=0,1,\dots,100$ ã«å¯ŸããŠïŒå蟺ã $x$ 軞ãŸã㯠$y$ 軞ã«å¹³è¡ãªé·æ¹åœ¢ã§ãã£ãŠïŒ$P$ ã®éã£ã $901$ åã®æ Œåç¹ã®ãã¡å¹³é¢ $z=k$ å
ã«å«ãŸãããã®å
šãŠããã®å
éšãŸãã¯åšäžã«å«ããã®ã®é¢ç©ã®æå°å€ã $S_k$ ãšããŸãïŒãã ãïŒ$z=k$ å
ã§ã® $P$ ã®éã£ãæ Œåç¹ãåäžçŽç·äžã«äžŠã¶ïŒãŸã㯠$1$ ç¹ã®ã¿ã§ãããšãïŒ$S_k=0$ ãšããŸãïŒ\
ãããã§ïŒäžé£ã®æäœã® **ã¹ã³ã¢** ã $S_0, S_1, \ldots, S_{100}$ ã®ç·ç©ã§å®ããŸãïŒæäœæ¹æ³ã¯å
šéšã§ $\dfrac{900!}{400!400!100!}$ éããããŸããïŒãããå
šãŠã«å¯Ÿããã¹ã³ã¢ã®ç·åã $M$ ãšããŸãïŒ
ããã®ãšãïŒ$M$ ãæ±ãïŒ$\gcd(M, 6^M)$ ã®æ£ã®çŽæ°ã®åæ°ã解çããŠãã ããïŒ
ã |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/7038 | A | OMCB011(A) | 100 | 337 | 360 | [
{
"content": "ã$MN \\parallel BC$ ã§ããïŒ$â MBC=108^\\circ \\div 2 =54^\\circ$ ã§ãããã $â BMN=180^\\circ - 54^\\circ=\\mathbf{126}^\\circ$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/7038"
}
] | ãæ£äºè§åœ¢ $ABCDE$ ã«ã€ããŠïŒç·å $AC$ ã®äžç¹ïŒç·å $BD$ ã®äžç¹ããããã $M,N$ ãšããŸãïŒ$â BMN$ ã®å€§ããã床æ°æ³ã§è§£çããŠäžããïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/4459 | B | OMCB011(B) | 100 | 329 | 354 | [
{
"content": "ã$(x+1)(x+2)(x+3)(x+4)(x+5)(x+6)$ ã®å±éãèããã°ïŒ$\\dfrac{6!}{x}$ ãæŽæ°ã«ãªã $x$ ãæ±ããã°ããïŒãããæºãã $x$ 㯠$6!=2^4 \\times 3^2 \\times 5$ ã®æ£ã®çŽæ°ã§ããããïŒæ±ããå€ã¯$$(2^0+2^1+2^2+2^3+2^4)(3^0+3^1+3^2)(5^0+5^1)=\\mathbf{2418}.$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/4459"
}
] | ã$\dfrac{(x+1)(x+2)(x+3)(x+4)(x+5)(x+6)}{x}$ ãæŽæ°ãšãªããããªæ£æŽæ° $x$ ã®ç·åãæ±ããŠãã ããïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/11081 | C | OMCB011(C) | 100 | 314 | 326 | [
{
"content": "ã$A$ ã®æ¬¡ã«éãç¹ã $P$ïŒãã®æ¬¡ã«éãç¹ã $Q$ ãšããïŒ$(P,Q)$ ã®å®ãæ¹ã¯ $3 \\times 2 = 6$ éãããïŒ$Q$ ãã $G$ ãžãšåããæ¹æ³ã¯ã¡ããã© $3$ éãååšããïŒå¯Ÿç§°æ§ãããããšã«æ³šæïŒïŒãããã£ãŠçã㯠$6 \\times 3 = \\mathbf{18}$ éãã§ããïŒ\r\n\r\n------\r\n**äŸïŒ**ããšãã° $(P,Q) = (B,C)$ ã®ãšãïŒ$C$ ãã $G$ ãžè¡ãæ¹æ³ã¯\r\n- $C \\to G$\r\n- $C \\to D \\to H \\to G$\r\n- $C \\to D \\to H \\to E \\to F \\to G$\r\n\r\nã® $3$ éãååšããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/11081"
},
{
"content": "ã$A$ ã®çŽåŸã«éãé ç¹ã $P$ ãšãïŒ$G$ ã®çŽåã«éãé ç¹ã $Q$ ãšãããšïŒå $P,Q$ ã«å¯ŸããŠçµè·¯ã $2$ éãååšããŸãïŒèããããçµè·¯ã¯ããã§å
šãŠãªã®ã§ïŒæ±ããå€ã¯ä»¥äžã®ããã«ãªããŸãïŒ\r\nã\r\n$$3 \\times 3 \\times 2 = \\mathbf{18}$$\r\n\r\n",
"text": "å¥è§£",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/11081/510"
}
] | ãç«æ¹äœ $ABCD - EFGH$ ã«ãããŠïŒç·å $AG$ ã¯ç«æ¹äœã®äœå¯Ÿè§ç·ïŒå
éšãéãæãé·ã察è§ç·ïŒã§ãïŒé ç¹ $A$ ãã蟺äžã®ã¿ãéã£ãŠé ç¹ $G$ ãŸã§éäžã§æ¥ãéãæ»ããã«ç§»åããæ¹æ³ã®ãã¡ïŒåãé ç¹ã $2$ å以äžééããªããã®ã¯ããã€ãããŸããïŒ\
ããã ãïŒã¹ã¿ãŒãå°ç¹ã® $A$ ããŽãŒã«å°ç¹ã® $G$ ãåºçºïŒå°çããæç¹ã§ééãããšã¿ãªããŸãïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/5182 | D | OMCB011(D) | 200 | 195 | 310 | [
{
"content": "ã$x \\lt y$ ãšããã° $x+y=13$ ãæºãã $(x,y)$ ã®çµã¯ $(1,12),(2,11),\\cdots ,(6,7)$ ã® $6$ åããïŒããããã«ã€ã㊠$(f(x),f(y))$ ã®çµã¿ãšããŠããåŸããã®ã¯ $(1,12),(2,11), \\cdots ,(12,1)$ ã® $12$ åååšããããæ±ããå€ã¯ $12^6=\\bf{2985984}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/5182"
}
] | ã$S=\lbrace1,2,3,\ldots ,12\rbrace$ ãšããŸãïŒä»¥äžã®æ¡ä»¶ãæºããé¢æ° $f\colon S\rightarrow S$ ã¯ããã€ãããŸããïŒ
- $x+y=13$ ãæºããå
šãŠã® $S$ ã®å
ã®çµ $(x,y)$ ã«ã€ã㊠$f(x)+f(y)=13$ ãæºããïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/6308 | E | OMCB011(E) | 200 | 312 | 318 | [
{
"content": "$$\\overline{ABCDE}\\times A=\\overline{EEEEEE}=3\\cdot7\\cdot11\\cdot13\\cdot37\\cdot E$$\r\nãã $11\\cdot 13\\cdot 37\\mid\\overline{ABCDE}\\cdot A$ ã ãïŒ$A$ 㯠$9$ 以äžã®æ£æŽæ°ãªã®ã§ $11,13,37$ ã®ãããã§ãå²ãåããªãïŒãã£ãŠ $11\\cdot 13\\cdot 37\\mid \\overline{ABCDE}$ ã§ããïŒæ¬¡ãæºããæ£æŽæ° $k$ ãååšããïŒ$$\\overline{ABCDE}=5291k\\quad (*)$$\r\n$10000\\lt\\overline{ABCDE}\\lt 99999$ ããïŒ$2\\leq k\\leq 19$ ãåŸãïŒãŸãïŒ$ ( * ) $ ã®å³èŸºã®$1$ã®äœã¯ $k$ ã® $1$ ã®äœã«çããã®ã§ïŒ$k=E$ ãŸã㯠$k=E+10$ ãæãç«ã€ïŒ$ ( * ) $ ãåé¡æã®çåŒã«ä»£å
¥ããŠæŽçããããšã§æ¬¡ãåŸãïŒ\r\n$$kA=21E$$\r\n- $k=E$ ã®ãšã\\\r\n$A=21$ ãåŸãïŒ$A$ ã $9$ 以äžã®æ£æŽæ°ã§ããããšã«åããïŒ\r\n- $k=E+10$ ã®ãšã\\\r\n$$(E+10)A=21E\\Longleftrightarrow(21-A)(E+10)=210$$\r\nãã $(A,E)=(6,4),(7,5)$ ãåŸãã®ã§ïŒããã«å¯Ÿå¿ã㊠$k=14,15$ ãšãªãïŒ$k=14$ ã®ãšã $\\overline{ABCDE}=74074$ ãšãªãäžé©ã§ããïŒäžæ¹ã§ $k=15$ã®ãšã$\\overline{ABCDE}=79365$ ãšãªãé©ããïŒ\r\n\r\n以äžãã $\\overline{ABCDE}=\\bf79365$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/6308"
},
{
"content": "ã$\\overline{ABCDE}=\\frac{E\\times 111\\times1001}{A}$ ã $1001$ ã®åæ°ãšãªããšãããšïŒ$\\overline{ABCDE}$ 㯠($2$ æ¡ã®æŽæ°) $\\times1001$ ã®åœ¢ãããŠããã¯ãã§ããïŒãã®æ $A=D, B=E$ ãšãªãäžé©ã§ãïŒ \r\nãããã£ãŠïŒ$\\frac{E\\times 111\\times1001}{A}$ 㯠$1001=7\\times 11\\times 13$ ã®åæ°ã§ãªãïŒããªãã¡ $A=7$ ãåãããŸã. \r\nãã®æïŒ$70000\\times 7=490000\\lt \\overline{EEEEEE}\\lt 560000=80000\\times 7$ ãã $E=5$ ãåãããŸãïŒ",
"text": "å¥è§£",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/6308/513"
}
] | ã$A,E$ 㯠$1$ ä»¥äž $9$ 以äžã®æŽæ°ïŒ$B,C,D$ 㯠$0$ ä»¥äž $9$ 以äžã®æŽæ°ã§ããïŒ$A,B,C,D,E$ ã¯çžç°ãªããŸãïŒ$$\overline{ABCDE}\times A=\overline{EEEEEE}$$ãæç«ããæïŒ$\overline{ABCDE}$ ã®å€ã¯äžæã«å®ãŸãã®ã§ïŒãã® $\overline{ABCDE}$ ã®å€ã解çããŠãã ããïŒ\
ããã ãïŒ$$\displaystyle\overline{a_na_{n-1}\cdots a_1a_0}=10^na_n+10^{n-1}a_{n-1}+\cdots+10a_1+a_0$$ ã§ãïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/10719 | F | OMCB011(F) | 200 | 246 | 281 | [
{
"content": "ãååŸ $5$ ã®è²çŽã¯å¿
ãèŠãããšãã§ãããïŒãã®ä»ã®è²çŽã¯èªåããã倧ããè²çŽã®äžã«ããã°èŠãããšãã§ããªãïŒãã£ãŠïŒååŸ $5$ ã®è²çŽã¯ãè¡šããè£ãã® $2$ éãïŒãã®ä»ã®è²çŽã¯ãè¡šããè£ããèŠããªããã® $3$ éãããïŒåè²çŽã®èŠãæ¹ãå®ãããšãïŒãããå®çŸãããéãæ¹ã¯ååšããïŒãããã£ãŠæ±ããèŠãæ¹ã¯ $2 \\times 3^4 = \\textbf{162}$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/10719"
}
] | ãæ°Žå¹³ãªæºãšïŒååŸããããã $1,2,3,4,5$ ã®åã®åœ¢ãããè²çŽã $1$ æãã€ããïŒ$i=1,2,3,4,5$ ã«å¯ŸããŠïŒååŸ $i$ ã®è²çŽã®è¡šé¢ã¯è² $2i-1$ ã§å¡ãããŠããïŒè£é¢ã¯è² $2i$ ã§å¡ãããŠããŸãïŒ
ãã ãïŒè² $1,2,...,10$ ã¯çžç°ãªããšããŸãïŒ\
ãæºã®äžã®å®ç¹ãå®ãïŒãã®äžã«èš $5$ æã®è²çŽãïŒå®ç¹ãšå
šãŠã®äžå¿ãäžèŽããããã«éããŸãïŒéããé çªïŒããã³ã©ã¡ãã®é¢ãäžã«ãããã¯ä»»æã§ãïŒéããç¶æ
ãçäžããèŠããšã $5$ æã®è²çŽã®**èŠãæ¹**ã¯äœéããããŸããïŒ\
ãéããé çªãç°ãªã£ããïŒé¢ã®åããéã£ããããªéãæ¹ã§ãã£ãŠãïŒçäžããèŠããšãã®èŠãæ¹ãåãã§ããã°ãããã®éãæ¹ã¯åºå¥ããŸããïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/6700 | G | OMCB011(G) | 300 | 154 | 193 | [
{
"content": "ã蟺 $AC, BC$ ã®äžç¹ããããã $D, M$ ãšãïŒçŽç· $BH$ ãš $AC$ ã®äº€ç¹ã $E$ ãšããïŒ \r\nã$|GD|=4a, |HM|=b$ ãšããïŒéå¿ã¯åé ç¹ãšå¯ŸèŸºã®äžç¹ã $2:1$ ã«å
åããç¹ã§ãããã $BD=12a$ ã§ããïŒãŸãïŒ$\\angle{CBH}=\\angle{GBH}$ ããïŒ\r\n$$BM=BG\\times \\frac{HM}{GH} = 2ab$$\r\nã§ããïŒããã«ïŒ\r\n$$\\angle CDB = 90^\\circ - \\angle GBH = 90^\\circ - \\angle CBH = \\angle BCD$$\r\nã§ããããäžè§åœ¢ $BCD$ 㯠$BC=BD$ ãªãäºç蟺äžè§åœ¢ã§ããã®ã§ïŒ$12a=4ab$ ã§ããïŒãã£ãŠ $b=3$ ã§ããïŒ\r\nåŸã£ãŠïŒäžè§åœ¢ ${GBM}$ ã«äžå¹³æ¹ã®å®çãçšããããšã§\r\n$$(2ab)^2 + (4+b)^2 = (8a)^2$$\r\nãã $a=\\dfrac{\\sqrt{7}}{2}$ ãåããïŒåŸã£ãŠïŒ\r\n$$AM=3GM=21,\\quad BC=4ab=6\\sqrt{7}$$\r\nãåããã®ã§ïŒäžè§åœ¢ $ABC$ ã®é¢ç©ã¯\r\n$$\\frac12\\times AM\\times BC=63\\sqrt{7}=\\sqrt{27783}$$\r\nã§ããïŒç¹ã«æ±ããçã㯠$\\mathbf{27783}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/6700"
},
{
"content": "ã$\\angle{HBC}=\\theta$ ãšãã. ãŸã蟺 $AC$ ã®äžç¹ã $M$ , $B$ ãã $AC$ ã«äžãããåç·ã®è¶³ã $K$ ãšãããš, $B,G,M$ ãš $B,H,K$ ã¯ããããå
±ç·ã§ãã. ãŸã, äžè§åœ¢ $BKC$ ã«ã€ã㊠$\\angle K=90^{\\circ} $ ãã, $(\\angle B=)\\angle C=90^{\\circ}-\\theta$. \\\r\nããã㧠äžè§åœ¢ $BMC$ ãš $BMA$ ã®é¢ç©ã«ã€ããŠä»¥äžãæç«ãã. \r\n$$\r\n\\begin{aligned}\r\n\\bigtriangleup BMC&=\\dfrac{1}{2}BM\\times BC \\sin{2\\theta}\\\\\\\\\r\n\\bigtriangleup BMA&=\\dfrac{1}{2}BM\\times BA \\sin{\\\\{(90^{\\circ}-\\theta)-2\\theta\\\\}}\r\n\\end{aligned}$$\r\n$BC=2BA\\cos{(90^{\\circ}-\\theta)}$ ããã³ $\t\\bigtriangleup BMC=\\bigtriangleup BMA$ ãã, 以äžãæç«.\r\n$$\r\n\\sin{(90^{\\circ}-3\\theta)}=2\\cos{(90^{\\circ}-\\theta)}\\sin{2\\theta}\r\n$$\r\nåè§ã®å
¬åŒãçšããŠæŽçãããš, $\\cos{\\theta}(8\\cos^2\\theta-7)=0$ ãæç«ã, $0^{\\circ}\\lt \\theta\\lt 90^{\\circ}$ ãã, $\\cos{\\theta}=\\sqrt{\\dfrac{7}{8}}$. ãã£ãŠ, $BC$ ã®äžç¹ã $N$ ãšã, $BN=x$ ãšãããš, $GH=GN-HN=x\\tan{2\\theta}-x\\tan{\\theta}$ ãã以äžãåŸã.\r\n$$\\Bigg(\\frac{\\sqrt{7}}{3}-\\frac{1}{\\sqrt{7}}\\Bigg)x=4 \\iff x=3\\sqrt{7}$$\r\nãã£ãŠ $\\bigtriangleup ABC=\\dfrac{1}{2}BC\\times AN=\\tan{\\angle B}\\times BN^2=\\sqrt{7}\\times(3\\sqrt{7})^2=\\sqrt{\\mathbf{27783}}$.",
"text": "äžè§æ¯ãçšãã解æ³",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/6700/512"
},
{
"content": "ãçŽç· $BH,BG$ ãš $AC$ ã®äº€ç¹ã $D,E$ ãšãããšïŒ$â DBC=â DBE$ ã〠$â BDC=90^{\\circ}$ ãªã®ã§ïŒ$BCE$ 㯠$BC=BE$ ãªãäºç蟺äžè§åœ¢ïŒç¹ã« $CD=DE$ ã§ããããïŒ$E$ ã $AC$ ã®äžç¹ãªããšãšäœµããã°\r\n$$AE : ED : DC = 2:1:1$$\r\nãåããïŒ\\\r\nããŸãïŒ$BC$ ã®äžç¹ã $M$ ãšããã°ïŒMenelausã®å®çãã\r\n$$\\frac{AD}{DC}Ã\\frac{CB}{BM}Ã\\frac{MH}{HA}=1 \\Rightarrow \\frac{MH}{HA} = \\frac{1}{6}$$\r\nããã« $AG:GM=2:1$ ãšäœµããããšã§ïŒ\r\n$$AG : GH : HM = 14:4:3$$\r\nãåŸãïŒåŸã£ãŠïŒ$AM=21, ~ HM=3$ ã§ããïŒ\r\n\r\nãè§åºŠèšç®ã«ããïŒ$\\triangle AMB \\sim \\triangle CMH$ ãåããããïŒ$AMÃMH=BMÃMC=63.$ $^{(*)}$ ãã£ãŠ $BC=2BM=6\\sqrt{7}$ ãªã®ã§ïŒæ±ããçã¯\r\n$$\\frac{1}{2}Ã6\\sqrt{7}Ã21=\\sqrt{\\mathbf{27783}}.$$\r\n\r\n---\r\n\r\n$(*)$ ããäžè¬ã«ïŒä»»æã®éè§äžè§åœ¢ $ABC$ ãšãã®åå¿ $H$ ã«ã€ããŠïŒ$AH$ ãš $BC$ ã®äº€ç¹ã $D$ ãšããã°ïŒ\r\n$$ADÃHD = BDÃDC$$\r\nãæãç«ã¡ãŸãïŒ$(\\triangle ADB \\sim \\triangle CDH$ ããïŒ$)$",
"text": "èšç®ç¡ãã§",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/6700/516"
}
] | ã$AB=AC,\angle BAC\lt 60^\circ$ ãªãäžè§åœ¢ $ABC$ ã®éå¿ã $G$ïŒåå¿ã $H$ ãšãããšããïŒ
$$\angle{CBH}=\angle{GBH},\quad GH=4$$
ãšãªããŸããïŒ
ãäžè§åœ¢ $ABC$ ã®é¢ç©ã®äºä¹ã¯æŽæ°ãšãªããŸãïŒãã®å€ãæ±ããŠãã ããïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/7115 | H | OMCB011(H) | 300 | 54 | 132 | [
{
"content": "$$a_n=(999+\\sqrt{999997})^n+(999-\\sqrt{999997})^n$$\r\nãšãããš $a_n$ ã¯æ£æŽæ°ã§ããïŒæ¬¡ã®æŒžååŒãæãç«ã€ïŒ\r\n$$a_{n+2}=1998a_{n+1}+1996a_n$$\r\nãã£ãŠæ¬¡ãåŸãïŒ\r\n$$a_0\\equiv 2,\\quad a_1\\equiv 8,\\quad a_{n+2}\\equiv 8a_{n+1}+6a_n\\quad\\pmod{10}$$\r\nãããçšãããš $a_1$ 以é㯠$10$ ãæ³ãšã㊠$8,6,6,4,8,8,2,4,4,6,2,2$ ãç¹°ãè¿ãããšããããïŒ\\\r\nãŸãïŒ$-1\\lt 999-\\sqrt{999997}\\lt 0$ ããïŒæ¬¡ããªããã€\r\n$$\\lfloor(999+\\sqrt{999997})^n\\rfloor=\r\n\\begin{cases}\r\na_n\\quad (n\\equiv 1\\pmod2)\\\\\\\\\r\na_n-1\\quad (n\\equiv 0\\pmod2)\r\n\\end{cases}$$\r\n以äžãã $\\lfloor(999+\\sqrt{999997})^n\\rfloor\\quad(n=1,2,...)$ 㯠$10$ ãæ³ãšã㊠$8,5,6,3,8,7,2,3,4,5,2,1$ ãç¹°ãè¿ãïŒãã£ãŠ $n$ ã®æ¡ä»¶ã¯æ¬¡ãšåå€ã§ããïŒ\r\n$$n\\equiv 2,10\\pmod{12}$$\r\næ±ããç·åã¯\r\n$$\\sum_{k=0}^{7}(12k+2+12k+10)+98=\\bf866.$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/7115"
},
{
"content": "todo $\\pmod{\\star}$ ãæããŠããå Žæã®ä¿®æ£ããã㯠$\\pmod{\\star}$ ããã¹ãŠç¥ããïŒ\r\n\r\n## ïœã¯ããã«ïœ\r\n\r\nã[å
¬åŒè§£èª¬](https:\\/\\/onlinemathcontest.com\\/contests\\/omcb011\\/editorial\\/7115) åæ§ã®èå¯ã¹ããããèžã¿, $10$ ã®å°äœã $5$ ãš $2$ ã®å°äœã«åããŠèããããšã§, \r\n$$a\\_0=2,\\quad a\\_1=3,\\quad a\\_{n+2}=3a\\_{n+1}+a\\_{n}\\pmod{5}$$\r\nãšå®ããæã« $a\\_{n}=1\\pmod {5}$ ãªã $n$ ã®æ±ºå®ãç®æšãšãªããŸã. \r\nãããš, å
¬åŒè§£èª¬ã«ãããããã«æ°åã®åšææ§ãå©çšãåŸçã®ããã§ã. \r\n\r\nãæ¬ãŠãŒã¶ãŒè§£èª¬ã§ã¯ãã®åšæã®çºèŠåã³ãã®åŸã®åŠçã楜ã«è¡ãæ¹æ³ã«ã€ããŠèª¬æããŸã. \r\n \r\n\r\n## ïœå
·äœçãªæ¹æ³ïœ\r\n\r\nã$b\\_n:=\\frac{a\\_{n+1}}{a\\_{n}}\\pmod{5}$ ãšããŸã. $b_n$ ã®å $4$ é
ãæçŽã«èšç®ããããš, $4,2,1,4$ ãšãªã£ãŠããŸã. \r\nããã§, $b\\_{n+1}$ 㯠$b\\_{n}$ ã®å€ã®ã¿ããå®ãŸããŸã. \r\nå®é, $a\\_{n+2}=3a\\_{n+1}+a\\_{n}=a\\_{n+1}(3+\\frac{1}{b\\_{n}})$ ãã, $b_{n+1}=3+\\frac{1}{b\\_n}\\pmod{5}$ ã§ã. \r\nããªãã¡, $b$ 㯠$4,2,1,4,2,1,4,2,1,\\dots$ ãšåšæ $3$ ã§ç¹°ãè¿ãããšãããããŸã. \r\n(泚: $b$ 㯠$a\\_{n}=0$ ãšãªããªããã, ä»åã®æ°å€èšå®ã®å
ã§å®çŸ©ã§ããŠããããã«æãããããããŸãã. ãããå®éã«ã¯ $b\\_n=0$ ãšãªãå Žåã§ãããçš®ã®æ£åœåãã§ããŸã. 詳现ã¯æ¬¡ç¯ãž)\r\n \r\n\r\nãä»,\r\n \r\n- $\\frac{a\\_{n+1}}{a\\_{n}}$ 㯠$4,2,1,4,2,1,4,\\dots$ ãšåšæ $3$ ã§ç¹°ãè¿ã. \r\n- $a\\_{3}=3a\\_{0} \\pmod{5}$ ã§ãã. \r\n \r\nãåãã£ãŠããŸã. \r\nããã§, åšææ§ãã $\\frac{a\\_{1}}{a\\_{0}}=\\frac{a\\_{4}}{a\\_{3}}$ ã§ããã®ã§, $a\\_{4}=3a\\_{1}\\pmod{5}$ ãåãããŸã. 以äžåæ§ã«ããŠ, $a\\_{n+3}=3a\\_{n}\\pmod{5}$ ãå
šãŠã® $n$ ã§æãç«ã¡ãŸã. \r\n\r\nã以äžãã, $a$ ã®å $3$ é
ã®å€ $(2,3,1)$ ãšäœµããŠ, \r\n$$a\\_{n}=\r\n\\begin{cases}\r\n2\\times 3\\^{m} &(n\\mod 3=0)\\\\\\\\\r\n3\\^{m+1} &(n\\mod 3=1)\\\\\\\\\r\n3\\^{m} &(n\\mod 3=2)\\\\\\\\\r\n\\end{cases}$$\r\n\r\nãšæ±ãŸããŸã. ãã ã, $m=\\left\\lfloor\\frac{n}{3}\\right\\rfloor$ ãšã, $a\\_{n}$ çŽåŸã®çå·ã¯$\\mod{5}$ ã§èããŠããŸã. \r\n\r\nãããã«, $3$ ã® $5$ ãæ³ãšããäœæ°ã $4$ ã§ããããšãçšãããš, 以äžãåãããŸã. \r\n\r\n- æ°å $a\\pmod{5}$ ã¯åšæ $12$ ã§ãã. \r\n- $n=0,1,\\dots,11$ ã®äžã« $a\\_{n}=1\\pmod{5}$ ãªã $n$ ã¯äžåºŠ $3$ ã€ãã, ããã $n$ ã® $3$ ã§ã®å°äœã¯å
šãŠç°ãªã. \r\n\r\nãããã«ããã«ä»åéœåã®è¯ãããšã«, $a_3=a_4=1$ ãæçŽèšç®ã®éã«æ¢ã«ç¥ã£ãŠãã, æ®ãã® $n=1\\pmod{3}, a\\_{n}=1\\pmod{5}$ ãæºãã $n$ ã«é¢ããŠã, $3\\^{4}=1\\pmod{5}$ ãã, $n=10$ ããã¡ã©ããã«åŸãããŸã. \r\n\r\nã 以äžãã $n=2,3,10\\pmod{12}$ ã®æã« $a\\_{n}=1\\pmod{5}$ ãšãªããšããçµè«ãåŸãããŸãã. \r\n\r\n## ïœè§£èª¬ã®è§£èª¬ïœ\r\n\r\nããã®ç¯ã§ã¯äžèšã®è§£èª¬ã«ã€ããŠå°ãã ãæ·±æãããŸã. \r\n#### åç¯ã®è§£æ³ã®æ¡åŒµ\r\nãŸãã¯åç¯ã® 泚 ã§è§Šãã, $a\\_n=0$ ãšãªãããå Žåã§ãåæ§ã®ææ³ã䜿ãããšããç¹ã«ã€ããŠèª¬æããŸã. \r\n\r\nä»åã¯, \r\n$$a\\_{0}=a\\_{1}=1, a\\_{n+2}=2a\\_{n}+3a\\_{n+1}\\pmod{5}$$\r\nã§å®ãŸãæ°åã®åšæãèããããšã«ããŸããã. \r\nãã®æ°åã¯, $(1,1,0,2,1,2,3,3,0,1,3,1,4,4,0,3,4,3,2,2,0,4,2,4,1,1,\\dots)$ ã®ããã«, $24$ é
㧠$1$ åšæãšãªããŸã. \r\n\r\n\r\nåç¯ã§ã¯, åšæãæ±ãããæ°åã®, é£æ¥é
ã®æ¯ã®å€ãæ°å $b$ ãšããŠããŸããã, ãããããšä»å㯠$\\frac{2}{0}$ ãåºãŠããŠããŸãã®ã§, 代ããã«, $[1:1], [1:0], [0:2]$ ã®ããã«æ¯ãã®ãã®ãå€ãšããŠæ±ãããšã«ããŸã. \r\nãã ã, $[0:0]$ ã¯èããªãããšãšã, $[x:y]$ ãš $[x\\^\\prime:y\\^\\prime]$ ã¯, \r\n$\\begin{cases}x=\\lambda x\\^\\prime\\pmod{5}\\\\\\\\y=\\lambda y\\^\\prime\\pmod{5}\\end{cases}$ ããšãã«æºãããã㪠$\\lambda=1,2,3,4$ ãååšããæã«, åãå€ãè¡šããšããŸã. \r\näŸãã°, $\\begin{cases}3=3\\times 1\\pmod{5}\\\\\\\\4=3\\times 3\\pmod{5}\\end{cases}$ ã§ããã®ã§, $[3:4]=[1:3]$ ã§ã. \r\nãã®ããã«ããããšã§, $[0:4]=[0:3]$ ã®ããã«æ¯ã®å€ã§ã¯æªå®çŸ©ãšãªãå Žåããã«ããŒã§ããŸã. \r\n\r\nããã®èšå·ã®å
ã§, æ¹ã㊠$b\\_{n}=[a\\_{n}:a\\_{n+1}]$ ãšå®ããåãèããŸã. \r\n$b\\_{n+1}=[a\\_{n+1}:a\\_{n+2}]=[a\\_{n+1}:2a\\_{n}+3a\\_{n+1}]$ ã§ã. \r\nã€ãŸã, $f([x:y])=[y:2x+3y]$ ãšããŠ, $b\\_{n+1}=b\\_{n}$ ã§ã. \r\n\r\n\r\nãããã§, [:] ã®åãæ±ãäžã§æ³šæãå¿
èŠãªç¹ã«è§ŠããŠãããŸã. \r\nä»åå®çŸ©ãã $f([x:y])=[y:2x+3y]$ ã§ãã, ã¡ãããšå®çŸ©åºæ¥ãŠããã®ã¯èªæãªäºã§ã¯ãããŸãã. \r\näŸãã°, $g([x:y])=[y:x\\^2]$ ãšãã $g$ ãå®çŸ©ã§ããŠãããèããŠã¿ãŸããã. \r\n$g([2:4])=[4:4]$ ã®ããã«, äžèŠã¡ãããšèšç®ã§ããããã§ã. ããã, $[2:4]=[1:2]$ ãšã衚瀺ã§ããããšãèãããš, $g([2:4])=g([1:2])=[2:1]$ ãšãèšç®ã§ããŠããŸããŸã. $[4:4]\\neq[2:1]$ ã§ããã®ã§, ããã¯ãã£ãŠã¯ãªããªãããšã§ã. ã€ãŸã, $g$ ã¯æ£ããå®çŸ©åºæ¥ãŠããŸãã. \r\näžæ¹, $f([x:y])=[y:2x+3y]$ ã®å Žå, $[x:y]=[\\lambda x:\\lambda y]$ ã®ãããªå¥ã®è¡šç€ºãããŠã, $f([\\lambda x:\\lambda y])=[\\lambda y:\\lambda (2x+3y)]=[y:2x+3y]$ ã®ããã«, äžè²«ããå€ãåŸãããŸã. (ãã®ããã«ã¡ãããšå®çŸ©ã§ããŠããããšã äžè¬ã« 'well-defined' ãšèšããŸã. ) \r\n\r\nããšããã, ä»åã®å Žåã§ã¯ $f([x:y])=[y:2x+3y]$ ãšããã°, $f(b\\_{n})=b\\_{n+1}$ ãšãªã, $b\\_{n+1}$ ã®å€ã $b\\_{n}$ ããã®ã¿èšç®ã§ãããšåãããŸãã. \r\n\r\nã$[x:y]$ ãšããŠåãããå€ã¯(èããŠããªã $[0:0]$ ãé€ãã°) $[0:1]$, $[1:0]$, $[1:1]$, $[1:2]$, $[1:3]$, $[1:4]$ ã® $6$ çš®é¡ã®ã¿ã§ã. ãã£ãŠ, $b$ ãšããŠåãããå€ãé«ã
$6$ éãã®ã¿ã§ããã®ã§, åšæ㯠$6$ 以äžãšãªããŸã. \r\nå®éã«, $b=(b\\_{0}=[1:1],[1:0],[0:1],[1:3],[1:2],[1:4],[1:1],\\dots)$ ãšåšæ $6$ ãšãªããŸã. \r\n\r\nããããŸã§ããã°, åŸã¯å
ã»ã©ã®å Žåãšåæ§ã«, $3a\\_{0}=a\\_{6}$ ã®èšç®ãš $b$ ã®åšæãã $a$ ã®åšæåã³åé
ã®å€ãæ¯èŒçç°¡åã«æ±ãŸããŸã. \r\n\r\n\r\n#### ä»åæ±ã£ãæŠå¿µã«ã€ããŠ\r\nãä»åæ±ã£ã $[x:y]$ ã®ããã«, ããã€ãã®å€ãçµã«ããŠ, åæåãåãå®æ°åã§äžèŽããæã«çããå€ãšã¿ãªã空éãå°åœ±ç©ºéãšèšããŸã. (éåžžå
šãŠ $0$ ã®çµã¯èããŸãã) \r\nä»åã¯æŽæ°ã« $\\mod 5$ äžã®æŒç®ãå
¥ããäžçã§ã®å°åœ±ç©ºé (èšå·ã§æžããš, $\\mathbb{F}\\_5\\mathbb{P}$ ã $\\mathbb{P}(\\mathbb{F}\\_5)$ ãšæžãããã¡) ã§æ°åã®åšæã®çºèŠã®ããã«å°å
¥ããŸããã, 競ææ°åŠã®äžçã§ã¯ãã®ä»ã«, åæåãå®æ°ãè€çŽ æ°ã®å Žåãªã©ãå°åœ±å¹ŸäœåŠãšåŒã°ãããžã£ã³ã«ã§çŸããããã§ã.",
"text": "åšææ§ã®çºèŠã«ã€ããŠ",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/7115/519"
},
{
"content": "ãæåãããããªã話ã§ãïŒ\r\n\r\n---\r\n\r\nã$\\alpha = 999+ \\sqrt{999997}, ~ \\beta = 999- \\sqrt{999997}$ ã«ã€ããŠïŒ$a_n = \\alpha^n + \\beta^n$ ã§ãïŒ\r\n\r\nãä»ïŒ$\\alpha,\\beta$ 㯠$x$ ã«ã€ããŠã® $2$ 次æ¹çšåŒ $x^2-1998x-1996=0$ ã®è§£ãªã®ã§ïŒãã¡ãã\r\n$$\\alpha^2-1998\\alpha-1996=0,ã\\beta^2-1998\\beta-1996=0$$\r\nã§ãïŒ$2$ åŒããããã« $\\alpha^n,\\beta^n$ ãæãããš\r\n$$\\alpha^{n+2}-1998\\alpha^{n+1}-1996 \\alpha^{n}=0,ã\\beta^{n+2}-1998\\beta^{n+1}-1996 \\beta^{n}=0$$\r\nãšãªãã®ã§ïŒãããã足ããŠ\r\n$$a_{n+2} - 1998a_{n+1} - 1996a_{n} = 0$$\r\nãåŸãŸãïŒ",
"text": "挞ååŒã®ç«ãŠæ¹",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/7115/525"
}
] | ã$(999+\sqrt{999997})^n$ ã $10$ é²æ³ã®å°æ°ã§è¡šãããšãã® $1$ ã®äœã $5$ ã§ãããããªïŒ$1$ ä»¥äž $100$ 以äžã®æŽæ° $n$ ã®ç·åãæ±ããŠãã ããïŒ\
ãäŸãã°ïŒå°æ° $7115.11$ ã® $1$ ã®äœã¯ $5$ ã§ãïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/5690 | A | OMC220(A) | 100 | 348 | 357 | [
{
"content": "$$\\angle ABP = \\angle PBC=30^{\\circ},\\quad AB = BC$$\r\nãæãç«ã€ããïŒäžè§åœ¢ $BPA$ ãš äžè§åœ¢ $BPC$ ã¯ååã§ããïŒãã£ãŠïŒ\r\n$$\\angle APC=2(180^\\circ - \\angle APB) = 2(180^\\circ - (180^\\circ - 12^\\circ - 30^\\circ))=\\textbf{84}^{\\circ}$$\r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc220/editorial/5690"
}
] | ãæ£äžè§åœ¢ $ABC$ ã«ãããŠïŒãã®å
éšã«ç¹ $P$ ããšããšïŒ
$$\angle BAP=12^{\circ}, \quad \angle ABP=30^{\circ}$$
ãæç«ããŸããïŒãã®ãšãïŒ$\angle APC$ ã®å€§ããã床æ°æ³ã§æ±ããŠãã ããïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/4571 | B | OMC220(B) | 300 | 248 | 294 | [
{
"content": "ã$10^6$ ã®äœãé©åã«è£ãããšã§ïŒåé¡ã¯ä»¥äžã®ããã«è¡šçŸã§ããïŒ\r\n\r\n- åæ¡ã®åã $6$ ã§ãã $7$ æ¡ä»¥äžã®æ£æŽæ°ãã¹ãŠã«ã€ããŠïŒãã®äž $6$ æ¡ã®ç·åãæ±ããïŒ\r\n\r\nãããŠïŒåæ¡ã®åã $6$ ã§ãã $7$ æ¡ä»¥äžã®æ£æŽæ°ã¯ïŒåºå¥ã®ãªã $6$ åã®çãåºå¥ã®ãã $7$ åã®ç®±ã«å
¥ããæ¹æ³ïŒç©ºã®ç®±ãèš±ãïŒã«å¯Ÿå¿ããããïŒãã®ç·æ°ã¯ ${}\\_{12}\\mathrm{C}\\_{6}$ åã§ããïŒããã ${}\\_{12}\\mathrm{C}\\_{6}$ åã®æ°ã®åæ¡ã®ç·å㯠${}\\_{12}\\mathrm{C}\\_{6}\\times 6$ ã§ããïŒåæ¡æ¯ã®ãã®ç·åãžã®å¯äžã¯åãã§ããããïŒããæ¡ã®ã¿ã«æ³šç®ãããšããã®ç·å㯠$({}\\_{12}\\mathrm{C}\\_{6}\\times 6)\\/7$ ãšãªãïŒä»¥äžããïŒæ±ããç·åã¯\r\n$$ \\frac{{}\\_{12}\\mathrm{C}\\_{6}\\times6}{7}\\times (10^5+10^4+\\cdots+10^0) = \\mathbf{87999912}$$\r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc220/editorial/4571"
},
{
"content": "以äžïŒ$6$ æ¡æªæºã®æ£æŽæ°ã¯å
é ã« $0$ ãä»ã足ã㊠$6$ æ¡ã®æ£æŽæ°ãšã¿ãªãïŒ\\\r\nãå $i\\~(0\\leq i\\leq 5)$ ã«ã€ããŠïŒåäœã®åã $n\\~(1\\leq n\\leq 6)$ ã〠$10^i$ ã®äœã $k\\~(0\\leq k\\leq n)$ ã§ãããã®ã®åæ°ã¯ïŒæ®ãã® $5$ æ¡ã« $n-k$ ãå²ãæ¯ãããšãèããã° $\\binom{n-k+4}{4}$ åã§ããïŒ\r\nãã£ãŠæ±ããå€ã¯æ¬¡ã®ããã«èšç®ã§ããïŒ\r\n$$\r\n\\begin{aligned}\r\n&\\sum_{n=1}^{6}\\sum_{k=0}^{n}\\binom{n-k+4}{4}\\cdot k(1+10+\\cdots+10^5)\\\\\\\\\r\n&=\\sum_{n=1}^{6}\\sum_{k=0}^{n-1}\\binom{k+4}{4}\\cdot (n-k)(1+10+\\cdots+10^5)\\\\\\\\\r\n&=\\sum_{n=1}^{6}\\left(n\\sum_{k=0}^{n-1}\\binom{k+4}{4}-5\\sum_{k=1}^{n-1}\\binom{k+4}{5}\\right)\\cdot(1+10+\\cdots+10^5)\\\\\\\\\r\n&=\\sum_{n=1}^{6}\\left(n\\binom{n+4}{5}-5\\binom{n+4}{6}\\right)\\cdot(1+10+\\cdots+10^5)\\\\\\\\\r\n&=\\sum_{n=1}^{6}\\binom{n+5}{6}\\cdot(1+10+\\cdots+10^5)\\\\\\\\\r\n&=\\binom{12}{7}\\cdot(1+10+\\cdots+10^5)\\\\\\\\\r\n&={\\bf 87999912}\r\n\\end{aligned}\r\n$$\r\n\r\nããã ãæ£æŽæ° $a,b\\~(a\\geq b)$ ã«å¯Ÿã次ãæãç«ã€ããšãçšããïŒ\r\n$$\\sum_{k=0}^{a}\\binom{k+b}{b}=\\sum_{k=1}^{a+1}\\binom{k+b-1}{b}=\\binom{a+b+1}{b+1}$$\r\nããã¯äž¡èŸºã« $\\binom{b}{b+1}=0$ ãå ãïŒäºé
ä¿æ°ã®æ§è³ª $\\binom{n}{k}+\\binom{n}{k+1}=\\binom{n+1}{k+1}$ ãç¹°ãè¿ãçšããã°ç€ºãããšãã§ããïŒ",
"text": "ãŽãªæŒãã®è§£æ³",
"url": "https://onlinemathcontest.com/contests/omc220/editorial/4571/511"
}
] | ã$10^6$ æªæºã®æ£æŽæ°ã®ãã¡ïŒåé²æ³è¡šèšã§åæ¡ã®åã $6$ 以äžã§ãããã®å
šãŠã«ã€ããŠïŒãã®ç·åãæ±ããŠãã ããïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/7581 | C | OMC220(C) | 300 | 203 | 276 | [
{
"content": "ã$M$ ã®èŠçŽ ãé ç¹ã«æã€ã°ã©ãã§ãã£ãŠïŒä»»æã® $M$ ã®èŠçŽ $x$ ã«å¯Ÿã㊠$x$ ãã $f(x)$ ã«åããŠæå¹èŸºã匵ãããŠãããã®ãèããïŒ\r\n\r\n----\r\n**è£é¡.**ã$f$ ãæ¡ä»¶ãæºããããšã¯ïŒåé ç¹ãé ç¹æ° $3$ ã®éè·¯ã«å«ãŸããããšãšåå€ã§ããïŒ\\\r\n**蚌æ.**ããŸãå¿
èŠæ§ã瀺ãïŒ$f$ ãæ¡ä»¶ãæºãããšãïŒ$f$ ã¯å
šå°ã§ãã $M$ ã¯æééåã§ããããïŒåé ç¹ã¯éè·¯ã«å«ãŸããïŒæ¬¡ã«ïŒ$f(f(f(x))) = x$ ããåéè·¯ã®é ç¹æ°ã¯ $3$ ã®çŽæ°ã§ãããïŒä»»æã® $M$ ã®èŠçŽ $x$ ã«å¯Ÿã㊠$f(x) \\neq x$ ã§ããããã©ã®éè·¯ã®é ç¹æ°ã $1$ ã§ã¯ãªãã®ã§ïŒå
šãŠã®éè·¯ã®é ç¹æ°ã¯ $3$ ã§ããïŒãã£ãŠç€ºãããïŒ\\\r\nã次ã«ååæ§ã瀺ãïŒä»»æã® $M$ ã®èŠçŽ $x$ ã¯å€§ãã $3$ ã®éè·¯ã«å«ãŸããã®ã§ïŒ$f(x) \\neq x$ ã§ããïŒããã« $f(f(f(x))) = x$ ã§ããïŒãã£ãŠç€ºãããïŒ\r\n----\r\n\r\nãè£é¡ããïŒ$N$ 㯠$M$ ã $3$ ã€ã®èŠçŽ ãããªãçµ $33$ åã«åãåã
ã«ã€ã㊠$2$ éãã®åããä»ããæ¹æ³ã®æ°ã«çããïŒãã®ãããªæ¹æ³ã¯ $\\dfrac{99!\\ 2^{33}}{33!\\ (3!)^{33}}$ éãããïŒããã $2$ ã§å²ãåããæ倧ã®åæ°ã¯Legendreã®å®çãã $\\bf64$ åã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc220/editorial/7581"
}
] | ã$M = \\{1,2,\ldots,99\\}$ ãšããŸãïŒ$f:M\to M$ ã§ãã£ãŠïŒä»»æã® $M$ ã®èŠçŽ $x$ ã«å¯ŸããŠ
$$f(x) \neq x,\quad f\big(f(f(x))\big) = x$$
ãæºãããã®ã®æ°ã $N$ ãšããŸãïŒ$N$ ã $2$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠäžããïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/7444 | D | OMC220(D) | 400 | 145 | 187 | [
{
"content": "ã$n=100$ ãšããïŒ\r\n$$S_1=\\sum_{k=1}^{n^2} \\bigg\\lfloor\\dfrac{k^2}{n^2}\\bigg\\rfloor,\\quad\r\nS_2=\\sum_{k=1}^{n^2}\\Big\\lfloor n\\sqrt{k}\\Big\\rfloor$$\r\nãšããïŒãã®ãšãïŒ$S_1$ 㯠$1\\le x\\le n^2$ ã®é åå
ã® $y=\\dfrac{x^2}{n^2}$ ãš $x$ 軞ã§å²ãŸããé åã«å«ãŸããæ Œåç¹ã®æ°ã«çããïŒ$S_2$ 㯠$1\\le x\\le n^2$ ã®é åå
ã® $y=n\\sqrt x$ ãš $x$ 軞ã§å²ãŸããé åã«å«ãŸããæ Œåç¹ã®æ°ã«çããïŒïŒãããã $x$ 軞äžã¯å«ãŸãïŒãã以å€ã®å¢çäžã®æ Œåç¹ã¯å«ãïŒ\\\r\nãããã§ïŒ$y=\\dfrac{x^2}{n^2}$ ãš $y=n\\sqrt x$ 㯠çŽç· $y=x$ ã§å¯Ÿç§°ã§ããããïŒ$S_1+S_2$ 㯠$1\\le x,y\\le n^2$ ã®é åå
ã®æ Œåç¹ã®æ°ãšïŒãã®é åå
ã® $y=n\\sqrt x$ äžã®æ Œåç¹ã®æ°ã®åã«çããïŒåŸã£ãŠïŒæ±ããç·å㯠$n^4 + n=\\bf{100000100}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc220/editorial/7444"
}
] | ã以äžã®å€ãæ±ããŠãã ãã.
$$\sum_{k=1}^{10000} \biggl( \bigg\lfloor\frac{k^2}{10000}\bigg\rfloor+\Big\lfloor 100\sqrt{k}\Big\rfloor \biggr)$$ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/6311 | E | OMC220(E) | 400 | 59 | 104 | [
{
"content": "ãçŽç· $HX$ ãšçŽç· $AB$ ã®äº€ç¹ã $Y$ ãšãïŒç·å $CY$ ã®äžç¹ã $M$ïŒ$M$ ã«ã€ã㊠$X$ ãšå¯Ÿç§°ãªç¹ã $Z$ ãšããïŒãã®ãšãïŒ\r\n$$\\angle XHC=\\angle AHC=180^{\\circ}-\\angle ABC=\\angle YBC$$\r\nãã $H, B, Y, C$ ã¯å
±åã§ããïŒãããã£ãŠäžè§åœ¢ $FHB$ ãš $FYC$ ã¯çžäŒŒãšãªãïŒãŸãïŒäžè§åœ¢ $XHB$ ãš $XCY$ïŒ$ZYC$ ã¯ãã¹ãŠçžäŒŒã§ããããïŒåè§åœ¢ $FHXB$ ãš $FYZC$ ã¯çžäŒŒïŒãããã£ãŠïŒ$FZ=22x, CZ=2x, YZ=23x$ ãšããïŒããã«äžç·å®çãã\r\n$$FX^2+FZ^2=2(FM^2+MZ^2)=2(CM^2+MZ^2)=ZC^2+ZY^2$$\r\nãæãç«ã€ïŒãã£ãŠïŒ$x=\\dfrac{22}{7}$ ã§ããïŒããŸïŒ$XY=ZC=2x$ ã§ããïŒãŸã $\\angle HYB=\\angle HCB=\\angle HAB$ ãã $HA=HY=23+2x=\\dfrac{205}{7}$ ã§ããïŒãã£ãŠè§£çãã¹ãå€ã¯ $\\textbf{212}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc220/editorial/6311"
},
{
"content": "å
¬åŒè§£èª¬ãšåãããã«ç¹ $Y$ ãåããŸãïŒ\r\n\r\nãŸãïŒ $X$ ãã $AB$ ã«äžãããåç·ã®è¶³ãç¹ $P$ ïŒ $\\tan \\angle BCF=\\dfrac{1}{x}$ ãšããã°ïŒ $YX=2x$ ãšæžãïŒ $FY$ ã $YP$ ã $XP$ ã $x$ ã§ãããããã®ã§ïŒçŽè§äžè§åœ¢ $FPX$ ã§ã®äžå¹³æ¹ã®å®ç㧠$x$ ã®äºæ¬¡æ¹çšåŒ(å®è³ªäžæ¬¡æ¹çšåŒ)ãç«ã¡ïŒããã解ã㊠$x= \\dfrac{22}{7}$ ãšæ±ãŸããŸãïŒããšã¯ $AH=YH=23+2x=\\dfrac{205}{7}$ ãšãªã£ãŠçããæ±ãŸããŸãïŒ",
"text": "ãŽãªæŒã",
"url": "https://onlinemathcontest.com/contests/omc220/editorial/6311/518"
}
] | ãäžè§åœ¢ $ABC$ ã®åå¿ã $H$ïŒ$C$ ãã蟺 $AB$ ã«äžãããåç·ã®è¶³ã $F$ ãšããŸãïŒèŸº $BC$ äžã« $\angle AHF=\angle XHF$ ãã¿ããç¹ $X$ ãååšãïŒ
$$BX=2, \quad FX=22, \quad HX=23$$
ãæãç«ã€ãšãïŒ$AH$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/5773 | F | OMC220(F) | 400 | 43 | 125 | [
{
"content": "ãã¹ç®ã®é ç¹ã®ãã¡ïŒäžãã $m$ çªç®ïŒå·Šãã $n$ çªç®ã®ãã®ã $(m, n)$ ã§è¡šãïŒ$1, 4$ ãæžã蟌ãŸãããã¹ãèµ€ãïŒ$2, 3$ ãæžã蟌ãŸãããã¹ãéãã¬ãïŒç°ãªãè²ã®å¢çã«ç·åãåŒãïŒãã®ãšãæ¡ä»¶ããïŒé£ãåã $2$ ãã¹ã«æžã蟌ãŸããæŽæ°ã®åã¯ïŒè²ãåãã§ããç®æã§ã¯ $5$ ã§ããïŒè²ãç°ãªãç®æã§ã¯ $5$ ã§ãªãïŒ\\\r\nãå¡ãæ¹ïŒæç·ã®åŒãæ¹ïŒã以äžã® $2$ éãã«åé¡ãã.\r\n- é·ã $2$ ã®æç·ã $2$ ã€åŒãããå Žå \\\r\né·ã $2$ ã®æç·ãšããŠãããããã®ã¯ $(1, n)$ ãš $(3, n)$ ãçµã¶ãã® ( $n=2, 3, \\ldots, 1468$ )ïŒ$(2, 1)$ ãš $(1, 2)$ ãŸã㯠$(3, 2)$ ãçµã¶ãã®ïŒ$(2, 1469)$ ãš $(1, 1468)$ ãŸã㯠$(3, 1468)$ ãçµã¶ãã®ã§ããïŒãããããéãªããªãããã« $2$ æ¬åŒãæ¹æ³ã¯ïŒ$(2,2)$ ãéã $3$ åãš $(2,1468)$ ãéã $3$ åãšãã以å€ã® $1465$ åã«åããŠèããããšã§ïŒ$3Ã3+2Ã3Ã1465+ {}\\_{1465}\\mathrm{C}\\_{2}$ éãååšãïŒæãç·ã®åŒãæ¹ã«ã€ããŠå¯Ÿå¿ããæŽæ°ã®æžã蟌ã¿æ¹ã¯ $2Ã2^3=16$ éãååšããããïŒæ¡ä»¶ãæºããæžã蟌ã¿æ¹ã¯ $17298864$ éãïŒ\r\n\r\n- é·ã $4$ ã®æç·ã $1$ ã€åŒãããå Žå \\\r\né·ã $4$ ã®æç·ãšããŠãããããã®ã¯ $(2, 1)$ ãš $(1, 4)$ ãŸã㯠$(3, 4)$ ãçµã¶ãã®ïŒ$(2, 1469)$ ãš $(1, 1466)$ ãŸã㯠$(3, 1466)$ ãçµã¶ãã®ïŒ$(1, n)$ ãŸã㯠$(3, n)$ ãš $(1, n+2)$ ãŸã㯠$(3, n+2)$ ãçµã¶ãã®ïŒ $n = 2, 3,\\ldots, 1466$ ïŒã§ããïŒãããã®æç·ã¯åèšã§ $2+2+4\\times1465$ éãååšãïŒæãç·ã®åŒãæ¹ã«ã€ããŠå¯Ÿå¿ããæŽæ°ã®æžã蟌ã¿æ¹ã¯ $2Ã2^2=8$ éãååšããããïŒæ¡ä»¶ãæºããæžã蟌ã¿æ¹ã¯ $46912$ éã. \r\n\r\nãã£ãŠïŒæ¡ä»¶ãæºããæŽæ°ã®æžã蟌ã¿æ¹ã¯ $\\textbf{17345776}$ éãã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc220/editorial/5773"
}
] | ã$2Ã1468$ ã®ãã¹ç®ããããŸãïŒãã®ãšãïŒåãã¹ã«ä»¥äžã®æ¡ä»¶ãæºããããã« $1, 2, 3, 4$ ã®æŽæ°ãæžã蟌ãæ¹æ³ã¯äœéããããŸããïŒ
- é£ããããã¹ã«ã¯ç°ãªãæŽæ°ãæžã蟌ãïŒ
- é£ããã $2$ ãã¹ã«æžã蟌ãŸããæŽæ°ã®åã $5$ ã§ãªããããªç®æã¯ã¡ããã© $4$ ç®æååšããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/8763 | A | OMCE003(A) | 300 | 108 | 162 | [
{
"content": "ã$P, Q, R, S$ ã®é ã§å€ãå®ããããšãèãããïŒ\\\r\nããŸã $P$ ã®éžã³æ¹ã¯ $1110$ éãããïŒããããã® $P$ ã«å¯Ÿãã $Q$ ã®å€ã®åè£ã¯\r\n$$P - 1000ïŒP - 100ïŒP - 10ïŒP + 110ïŒP + 1010ïŒP + 1100$$\r\nã® $6$ ã€ã§ãããïŒãã®ãã¡ $1$ ä»¥äž $1110$ 以äžã®ç¯å²ã«å«ãŸãããã®ã¯ïŒ$P$ ã®éžã³æ¹ã«ãããïŒã¡ããã© $3$ ã€ã§ããïŒ\r\n\r\n<details><summary>çç±<\\/summary>\r\n$$\\begin{aligned}\r\nQ_1 &= P - 1000, &Q_2 &= P - 100, &Q_3 &= P - 10 \\\\\\\\\r\nQ_4 &= P + 110, &Q_5 &= P + 1010, &Q_6 &= P + 1100\r\n\\end{aligned}$$\r\nãšããïŒããŸïŒ\r\n$$ 1000 + 110 = 100 + 1010 = 10 + 1100 = 1110 $$\r\nã§ããããšã«æ³šæãããšïŒ\r\n$$\\begin{aligned}\r\nQ &= Q_4, Q_5, Q_6 &&(1 \\leq P \\leq 10) \\\\\\\\\r\nQ &= Q_3, Q_4, Q_5 &&(11 \\leq P \\leq 100) \\\\\\\\\r\nQ &= Q_2, Q_3, Q_4 &&(101 \\leq P \\leq 1000) \\\\\\\\\r\nQ &= Q_1, Q_2, Q_3 &&(1001 \\leq P \\leq 1110) \r\n\\end{aligned}$$\r\nããããããªãããïŒ$Q$ ãšããŠããããå€ã¯ã¡ããã© $3$ ã€ãšãªãïŒ\r\n<\\/details>\r\n\r\nããã£ãŠ $Q$ ã®éžã³æ¹ã¯ $3$ éãååšããïŒæ®ãã® $2$ æ°ã«ã€ããŠãåæ§ã«éžã³æ¹ã $3$ éããã€ååšããããšãåããããïŒæ¡ä»¶ãã¿ããçµã®åæ°ã¯å
šéšã§ $1110 \\times 3^3 = \\mathbf{29970}$ åã§ããïŒ\r\n\r\n------\r\n\r\n **å¥è§£ïŒ** $S, R, Q, P$ ã®é ã« $4$ æ°ãå®ããããšã§ãïŒå€åã®æ£è² ãå€ããã®ã¿ã§ã»ãšãã©åæ§ã«è§£ãããšãã§ããïŒã©ã®ãããªå®ãæ¹ãããã«ããïŒ$1110$ ãæ³ãšããŠèããïŒ$n$ ãã $110$ ãåŒãããšã¯ïŒ$\\mathrm{mod} ~ 1110$ ã§ã¯ $1000$ ã足ãããšãšçããïŒããšãéèŠã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/8763"
}
] | ã$1$ ä»¥äž $1110$ 以äžã®æŽæ°ã®çµ $(P, Q, R, S)$ ã§ãã£ãŠïŒä»¥äžãã¿ãããã®ã¯å
šéšã§ããã€ãããŸããïŒ
- $P - Q, Q - R, R - S$ ã¯ãããã
$$-1100, -1010, -110, 10, 100, 1000$$
ã®ããããã«çããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/8320 | B | OMCE003(B) | 400 | 62 | 91 | [
{
"content": "ãäžè§åœ¢ $ABM$ ã®å€æ¥åã $\\Omega$ ãšãããšãïŒåçŽç· $BD$ ãš $\\Omega$ ãç¹ $B$ 以å€ã«å
±æç¹ã $1$ ã€ãã€ã®ã§ïŒããã $E$ ãšããïŒãããšååšè§ã®å®çãã\r\n$$\\angle AME = \\angle MAE = 60^{\\circ}$$\r\nãããããã®ã§ïŒäžè§åœ¢ $AME$ ã¯æ£äžè§åœ¢ã§ããïŒããã§æ¬¡ã®è£é¡ãæãç«ã€ïŒ\r\n\r\n---\r\n\r\n**è£é¡ïŒ**$E$ ã¯ç·å $BD$ïŒäž¡ç«¯ãé€ãïŒäžã®ç¹ã§ããïŒ\r\n\r\n<details><summary>è£é¡ã®èšŒæ<\\/summary>\r\nã$BE \\geq BD$ ã§ãããšä»®å®ããïŒç¹ $A$ ããçŽç· $BD, BC$ ã«ããããåç·ã®è¶³ããããã $H_1, H_2$ ãšããïŒ\r\n$$\\angle ADB = \\angle MAC \\lt \\angle BAC = 60^{\\circ} - \\angle ACB$$\r\nã§ããããšãã $\\angle ADB$ ã¯éè§ãªã®ã§ïŒ$H_1$ ã¯åçŽç· $DB$ äžã«ããïŒãã£ãŠ $H_1E \\geq H_1D$ ãããããïŒ$AE \\geq AD$ ãåŸãïŒåæ§ã« $\\angle AMB$ ãéè§ã§ããããšãã $H_2$ ã¯åçŽç· $MB$ äžã®ç¹ãªã®ã§ïŒ $H_2C \\gt H_2M$ ãã $AC \\gt AM$ ãåŸãïŒãããã®ããšã« $AE = AM$ ãããããããšã§ $AC \\gt AD$ ãåŸããïŒããã¯æ¡ä»¶ $AC : AD = 16 : 19$ ã«ççŸïŒããã« $BE \\lt BD$ ãªã®ã§ïŒäž»åŒµã¯æãç«ã€ïŒ\r\n<\\/details>\r\n\r\n---\r\n\r\nãååšè§ã®å®çãã $\\angle AMB = \\angle AEB$ ã§ããïŒ$\\angle AMC = \\angle DEA$ ãåŸãããïŒãããš $\\angle MAC = \\angle EDA$ ãã $\\triangle AMC \\sim \\triangle DEA$ ããããïŒãã®çžäŒŒæ¯ã¯ $16 : 19$ ã§ããïŒããã« $AM = AE, BM = CM$ ãããããããšã§ $BM : AM = 16 : 19$ ãåŸããïŒããã«äžè§åœ¢ $ABM$ ã«ã€ããŠäœåŒŠå®çãé©çšãããªã©ããã° $AB : BM : AM = 5 : 16 : 19$ ããããïŒãã£ãŠïŒæ¯ã®èšç®ã«ãã以äžã®ããã«é 次é·ããæ±ããããïŒ\r\n$$BM = CM = \\frac{16}{5}ïŒAM = AE = \\frac{19}{5}ïŒDE = \\frac{361}{80}$$\r\nãŸãïŒ$4$ ç¹ $A, B, M, E$ ãå
±åãã€äžè§åœ¢ $AME$ ãæ£äžè§åœ¢ã§ããããšãã $BE = AB + BM$ ãæãç«ã€ã®ã§ïŒ\r\n$$BD = BE + DE = 1 + \\frac{16}{5} + \\frac{361}{80} = \\frac{697}{80}$$\r\nãåŸãïŒããã«ïŒè§£çãã¹ãå€ã¯ $\\mathbf{777}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/8320"
},
{
"content": "ã$\\angle ABD=\\angle CBD=60^{\\circ}$ ãšïŒéåžžã«æ±ããããè§ãç¹ $B$ ã®åšãã«éãŸã£ãŠããã®ã§ïŒç¹ $B$ ãåç¹ã«ãšã£ãŠåº§æšãèšå®ããŠã¿ããïŒ\r\n\r\nã$A \\left( \\dfrac{1}{2}, \\dfrac{\\sqrt{3}}{2}\\right), M(x, -\\sqrt{3}x), C(2x, -2\\sqrt{3}x), D(t,0)$ ãšããïŒ\\\r\nã$AC^2=16x^2+4x+1,AD^2=t^2-t+1$ ãã \r\n$$\\tag{1} 16^2(t^2-t+1)=19^2(16x^2+4x+1) $$\r\nã$\\overrightarrow{AM} \\cdot \\overrightarrow{AC}=8x^2+3x+1, \\overrightarrow{DA} \\cdot \\overrightarrow{DB}=t(t-\\frac{1}{2})$ ãçšããŠïŒ$\\angle MAC,\\angle ADB$ ã®äœåŒŠ ($\\cos$) ã®äºä¹ãæ±ãããš\r\n$$\\dfrac{(8x^2+3x+1)^2}{(4x^2+2x+1)(16x^2+4x+1)}=\\dfrac{(t-\\frac{1}{2})^2}{t^2-t+1}$$\r\nãåŸãïŒäžèŠ $4$ 次åŒã«ãªãããã ãïŒæ¬¡ã®ããã«åŒå€åœ¢ããã°ããïŒ\r\n$$1-\\dfrac{3x^2}{(4x^2+2x+1)(16x^2+4x+1)}=1-\\dfrac{\\frac{3}{4}}{t^2-t+1}$$\r\nãåŒ $(1)$ ãçšããã° $x$ ã«ã€ããŠã® $2$ 次æ¹çšåŒãšãªãïŒ$x=\\dfrac{8}{5}$ ãåŸãïŒæåŸã«ããäžåºŠåŒ $(1)$ ãçšã㊠$t$ ãæ±ããã°ããïŒ",
"text": "座æšãçšããæ¹æ³ïŒéæšå¥šïŒ",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/8320/499"
},
{
"content": "#### ã€ã³ãã\r\nã$BD$ ãš $AM, AC$ ã®äº€ç¹ããããã $X, Y$ ãšããŸãïŒãŸãçŽç· $BC$ äžã« $Z$ ã $\\triangle ABZ$ ãæ£äžè§åœ¢ã«ãªãããã«ãšããŸã (ãªãïŒ$Z$ ã¯äžè§åœ¢ã®å
è§ã®äºçåç·ã®é·ããæ±ããããã ãã«çšããã®ã§ïŒãã®å
¬åŒãç¥ã£ãŠããæ¹ã¯ãšãå¿
èŠã¯ãããŸãã)ïŒãŸãïŒ$BM=CM=x$ ãšããŸãïŒ\\\r\nå³ãå®éã«æžããŠã¿ããšïŒ$\\triangle XYA\\sim\\triangle XAD$ ãèŠããã®ã§ãã®çžäŒŒããç«åŒããŠçããæ±ããŠãããŸãããïŒ\r\n---\r\n#### å®éã«è§£ã\r\nã$\\triangle ABM$ ã«ãããŠäœåŒŠå®çã»è§ã®äºçåç·ã®æ§è³ªãã $AX$ ãæ±ãŸãïŒ$\\triangle AZM\\sim\\triangle XBM$ åã³ $\\triangle AZC\\sim\\triangle YBC$ ã«é¢ããŠã®çžäŒŒæ¯ãã $XB, YB$ ãåããã®ã§ïŒ$XY$ ãæ±ããããŸãïŒãŸãïŒ$\\triangle ABC$ ã«é¢ããŠè§ã®äºçåç·ã®æ§è³ªãã $YA:AD$ïŒã€ãŸã $\\triangle XYA\\sim\\triangle XAD$ ã®çžäŒŒæ¯ãèšç®ã§ããŸãïŒ ãããã£ãŠ $XY:XA=YA:AD$ ã«ãã£ãŠåŸãããçåŒ $XY\\cdot AD=XA\\cdot YA$ ã $x$ ã«é¢ããæ¹çšåŒãšããŠç«ãŠãããšãã§ããŸãïŒããšã¯ããã解ãã°è¯ãã§ãïŒ\r\n<details><summary>å
·äœçãªèšç®<\\/summary>\r\n$$AX=\\frac1{1+x}AM=\\frac{\\sqrt{x^2+x+1}}{x+1}$$\r\n$$BX=ZA\\cdot\\frac{BM}{ZM}=\\frac{x}{x+1}$$\r\n$$BY=ZA\\cdot\\frac{BC}{ZC}=\\frac{2x}{2x+1}$$\r\n$$XY=BY-BX=\\frac{x}{(x+1)(2x+1)}$$\r\n$$YA:AD=\\frac{16}{2x+1}:19$$\r\nãšãªãã®ã§ïŒ$XY:XA=YA:AD$ ãã\r\n$$\\frac{x}{(x+1)(2x+1)}:\\frac{\\sqrt{x^2+x+1}}{x+1}=\\frac{16}{2x+1}:19\r\n\\Longleftrightarrow 19x=16\\sqrt{x^2+x+1}$$\r\n($x\\gt0$ ã§ããããšã«æ³šæããŠãã ãã) ãšããããã«è¡šåŒãåŸãããšãã§ããŸãïŒ\\\r\nããã解ããš $x=\\dfrac{16}5$ ãåŸãããŸãïŒã©ã®èŸºãèšç®ã§ããããã«ãªã£ãã®ã§ïŒããšã¯ã©ã®æ¹éã§ãçããæ±ãŸãã§ãããïŒ\\\r\näŸãã°ä»¥äžã®ããã«èšç®ããããšãã§ããŸãïŒ\r\n$$BD=BX+XD=\\frac{x}{x+1}+AX\\cdot\\frac{AD}{YA}=\\frac{16x+19(2x+1)\\sqrt{x^2+x+1}}{16(x+1)}=\\frac{697}{80}$$\r\n<\\/details>",
"text": "é·ã远跡",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/8320/505"
},
{
"content": "åçŽç· $BC$ ãäžè§åœ¢ $ABD$ ã®å€æ¥åãšåã³äº€ããç¹ã $F$ ãšããïŒäžè§åœ¢ $ADF$ ãæ£äžè§åœ¢ã§ããããšãååšè§ã®å®çããããã«ãããïŒ\\\r\näžè§åœ¢ $CMA,AMF$ ã®çžäŒŒãããã(ååšè§ã®å®ç)ã®ã§ïŒ$$BM:MA=CM:MA=CA:AF=16:19$$\r\näœåŒŠå®ç㧠$BM=16\\/5$ ãšãããïŒ\\\r\nãã¬ããŒã®å®çã§äž¡èŸºãæ£äžè§åœ¢ $ADF$ ã®äžèŸºã®é·ãã§å²ãããšã§\r\n$$BD=AB+BF=AB+BM+MF=1+\\frac{16}{5}+\\frac{16}{5}\\cdot \\frac{19}{16}\\cdot \\frac{19}{16}=\\frac{697}{80}$$\r\nãšãªãã®ã§æ±ããã¹ãå€ã¯ $697+80=\\mathbf{777}$ïŒ",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/8320/514"
}
] | ãåžåè§åœ¢ $ABCD$ ãäžããããŠããïŒèŸº $BC$ ã®äžç¹ã $M$ ãšãããšããïŒ
$$AB = 1ïŒAC : AD = 16 : 19ïŒ \\\\
\angle ABD = \angle CBD = 60^{\circ}ïŒ\angle MAC = \angle ADB$$
ãæãç«ã¡ãŸããïŒãã®ãšãç·å $BD$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a + b$ ã®å€ã解çããŠãã ããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/11150 | C | OMCE003(C) | 400 | 116 | 148 | [
{
"content": "ãæ Œåç¹ã§ãã£ãŠ $x, y$ 座æšããšãã« $3$ ã®åæ°ãšãªããã®ã**çœãç¹**ãšåŒã³ïŒ$x, y$ 座æšããããã $3$ ã§å²ã£ããšãïŒäžæ¹ãäœã $1$ ã§ããäžæ¹ãäœã $2$ ãšãªããã®ã**é»ãç¹**ãšåŒã¶ïŒä»»æã® $1 \\leq n \\leq 36$ ãªãæŽæ° $n$ ã«å¯Ÿã $P_n$ ã® $x, y$ 座æšã®å㯠$n$ ãªã®ã§ïŒOMCåã®ç§»åãç¹°ãè¿ãæ¹æ³ã«ããã $P_3, P_6, ..., P_{36}$ ã® $12$ åã¯å¿
ãçœãç¹ãé»ãç¹ã§ããïŒéã«ïŒ$P_1, P_2, ..., P_{36}$ ã®ãã¡å
ã»ã©ã® $12$ å以å€ã® $24$ åã¯ïŒå¿
ãçœãç¹ã§ãé»ãç¹ã§ããªãïŒ$P_3, P_6, ..., P_{36}$ ã®ãã¡ã¡ããã© $1$ åãçœãç¹ãšãªããããªç§»åã®ç¹°ãè¿ããæ°ããã°ããïŒãŸãïŒ$1$ åã®ç§»åã®ãã¡ $x, y$ 軞ã®æ£ã®æ¹åã«ç§»åããããšãïŒãããããå³ããäžããšè¡šãããšã«ããïŒ \\\r\nããŸãã¯çœãç¹ãèµ·ç¹ãšã移åã $3$ åç¹°ãè¿ãããšãïŒç§»ååŸã«çœãç¹ã»é»ãç¹ã«å°éãã移åæ¹æ³ãããããæ°ãããïŒçœãç¹ã«å°éããã®ã¯ãå³ã« $3$ åé£ç¶ç§»åããŸãã¯ãäžã« $3$ åé£ç¶ç§»åãã® $2$ éãã§ããïŒé»ãç¹ã«å°éããã®ã¯ãã以å€ã® $2^3 - 2 = 6$ éãã§ããïŒ \\\r\nã次ã«é»ãç¹ãèµ·ç¹ãšã移åã $3$ åç¹°ãè¿ãããšãïŒç§»ååŸã«çœãç¹ã»é»ãç¹ã«å°éãã移åæ¹æ³ãããããæ°ãããïŒå°éãåŸã $4$ ã€ã®ç¹ã®ãã¡çœãç¹ã¯ã¡ããã© $1$ ã€ã§ããïŒãªããã€ããã¯ãå³ã« $1$ïŒäžã« $2$ 移åããå
ãããå³ã« $2$ïŒäžã« $1$ 移åããå
ãã«äœçœ®ããïŒã©ã¡ãã®å Žåã«ããçœãç¹ã«å°éããã®ã¯ ${}\\_{3}\\mathrm{C}\\_{1} = 3$ éãã§ããïŒé»ãç¹ã«å°éããã®ã¯ $2^3 - 3 = 5$ éãã§ããïŒ \\\r\nã以åŸïŒ$P_0$ ã¯åç¹ã§ãããšããïŒ$P_0$ ã¯çœãç¹ã§ããïŒïŒãŸãïŒ$P_0$ ãã $P_{36}$ ãŸã§ã®ç§»åã®ç¹°ãè¿ããïŒã$P_0$ ãã $P_3$ãã$P_3$ ãã $P_6$ãã»ã»ã»ã$P_{33}$ ãã $P_{36}$ãã® $12$ åã®ç§»åã«åå²ããŠèããïŒãã® $12$ åã«å«ãŸãããçœãç¹ããçœãç¹ããçœãç¹ããé»ãç¹ããé»ãç¹ããçœãç¹ããé»ãç¹ããé»ãç¹ãã«ç§»åããåæ°ããããã $(W_w, W_b, B_w, B_b)$ ãšè¡šãïŒ$P_3, P_6, ..., P_{36}$ ã®ãã¡ã©ããçœãç¹ãã決ãŸã£ãŠããã°ïŒãã®æ¡ä»¶ã®ããšã§ç§»åãç¹°ãè¿ãæ¹æ³ã®åæ°ã¯\r\n$$2^{W_w} \\times 6^{W_b} \\times 3^{B_w} \\times 5^{B_b}$$\r\n\r\nãšè¡šããïŒ$P_3, P_6, ..., P_{36}$ ã®ãã¡çœãç¹ãšãªããã®ã®éžã³æ¹ãã¹ãŠã«å¯Ÿãããã®å€ã®ç·åãæ±ããã°ããïŒ $(W_w, W_b, B_w, B_b)$ ã¯ä»¥äžã® $3$ éããããåŸãïŒ\r\n- çœãç¹ã $P_3$ ã®ãšãïŒ$(W_w, W_b, B_w, B_b) = (1, 1, 0, 10)$ ã§ããïŒ\r\n- çœãç¹ã $P_{36}$ ã®ãšãïŒ$(W_w, W_b, B_w, B_b) = (0, 1, 1, 10)$ ã§ããïŒ\r\n- çœãç¹ã $P_6, P_9, ..., P_{33}$ ã®ããããã§ãããšãïŒ$(W_w, W_b, B_w, B_b) = (0, 2, 1, 9)$ ã§ããïŒ\r\n\r\nç¹ã« $3$ çªç®ã®ã±ãŒã¹ã ãšçœãç¹ã®åè£ã $10$ åããããšã«æ³šæããã°ïŒæ±ããåæ°ã¯æ¬¡ã®ããã«èšç®ã§ããïŒ\r\n$$2 \\times 6 \\times 5^{10} + 6 \\times 3 \\times 5^{10} + 6^2 \\times 3 \\times 5^9 \\times 10 = \\mathbf{2402343750}$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/11150"
},
{
"content": "ãå
¬åŒè§£èª¬ãšåæ§ã«ïŒ$x,y$ 座æšããšãã« $3$ ã®åæ°ã§ãããããªç¹ã**çœãç¹**ãšåŒã¶ïŒ\\\r\nã移åã $3k$ åç¹°ãè¿ãããšãã«ïŒ\r\n- $P_1, \\cdots, P_{3k}$ ã®ãã¡çœãç¹ãååšããªãå Žåã®æ°ã $a_k$\r\n- $P_1, \\cdots, P_{3k-1}$ ã¯ããããçœãç¹ã§ãªãïŒ$P_{3k}$ ãçœãç¹ã§ãããããªå Žåã®æ°ã $b_k$\r\n- $P_1, \\cdots, P_{3k}$ ã®ãã¡ã¡ããã©äžã€ãçœãç¹ã§ããïŒãã㯠$P_{3k}$ ã§ã¯ãªããããªå Žåã®æ°ã $c_k$\r\n\r\nãšããïŒãããã¯äºãã«æåã§ããïŒããŸæ±ããããã®ã¯ $b_{12}+c_{12}$ ã§ããïŒ\\\r\nãå
¬åŒè§£èª¬ãšåæ§ã«ïŒãããã®å Žåã®é·ç§»ãèãããšïŒæ¬¡ã®æŒžååŒãåŸãïŒ\r\n$$\\begin{aligned}\r\n a_{n+1} &= 5a_n \\\\\\\\\r\n b_{n+1} &= 3a_n \\\\\\\\\r\n c_{n+1} &= 6b_n+5c_n\r\n \\end{aligned}$$\r\nãããšã¯ $a_1=6, b_1=2, c_1=0$ ãçšããŠæŒžååŒã解ãã°ããïŒ$b_n=18Ã5^{n-2}\\ \\ (nâ§2)$ ã¯å®¹æã«ãããã®ã§ïŒ\r\n$$c_{n+1}=5c_n+108Ã5^{n-2}, c_2=12$$\r\n ã解ãã°ããããšããããïŒããã解ãã° $c_n=(108n-156)Ã5^{n-3}\\ \\ (nâ§2)$ ãšãªãïŒæ±ããã¹ãå€ã¯\r\n$$b_{12}+c_{12}=18Ã5^{10}+1140Ã5^9=246Ã5^{10}=\\mathbf{2402343750}$$",
"text": "挞ååŒãäœã£ãŠè§£ã",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/11150/500"
},
{
"content": "çµè«ããèšããšçãã¯ãã®å€ã«ãªããŸã\r\n\r\n$$\\begin{pmatrix}0&1&0&2\\\\\\\\\\end{pmatrix}\\begin{pmatrix}0&0&0&0\\\\\\\\2&0&6&0\\\\\\\\3&0&5&0\\\\\\\\0&3&0&5\\end{pmatrix}^{12}\\begin{pmatrix}1\\\\\\\\0\\\\\\\\0\\\\\\\\0\\\\\\\\\\end{pmatrix}$$\r\n\r\n# 蚌æ\r\n\r\n$P_{3n}$ã®æç¹ã«ãŠã\r\n- $(x_{3n}\\mod{3},y_{3n}\\mod{3})=(0,0)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®ã$a_n$\r\n- $(x_{3n}\\mod{3},y_{3n}\\mod{3})=(0,0)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®ã$b_n$\r\n- $(x_{3n}\\mod{3},y_{3n}\\mod{3})= (1,2)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®ã$c_n$\r\n- $(x_{3n}\\mod{3},y_{3n}\\mod{3})= (1,2)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®ã$d_n$\r\n\r\nãšããã$(a_0,b_0,c_0,d_0)=(1,0,0,0)$ã§ããã\r\n\r\nãã®ãšãã\r\n- $(x_{3n}\\mod{3},y_{3n}\\mod{3})= (2,1)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®$=c_n$\r\n- $(x_{3n}\\mod{3},y_{3n}\\mod{3})= (2,1)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=d_n$\r\n\r\nãæãç«ã€ã\r\n\r\n\r\nãã®ãšãã\r\n\r\n- $(x_{3n+1}\\mod{3},y_{3n+1}\\mod{3})= (0,1)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®$=a_n+c_n$\r\n- $(x_{3n+1}\\mod{3},y_{3n+1}\\mod{3})= (1,0)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®$=a_n+c_n$\r\n- $(x_{3n+1}\\mod{3},y_{3n+1}\\mod{3})= (2,2)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®$=2c_n$\r\n- $(x_{3n+1}\\mod{3},y_{3n+1}\\mod{3})= (0,1)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=b_n+d_n$\r\n- $(x_{3n+1}\\mod{3},y_{3n+1}\\mod{3})= (1,0)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=b_n+d_n$\r\n- $(x_{3n+1}\\mod{3},y_{3n+1}\\mod{3})= (2,2)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=2d_n$\r\n\r\nã§ããã\r\n\r\n- $(x_{3n+2}\\mod{3},y_{3n+2}\\mod{3})= (0,2)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®$=a_n+3c_n$\r\n- $(x_{3n+2}\\mod{3},y_{3n+2}\\mod{3})= (0,2)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=b_n+3d_n$\r\n- $(x_{3n+2}\\mod{3},y_{3n+2}\\mod{3})= (2,0)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®$=a_n+3c_n$\r\n- $(x_{3n+2}\\mod{3},y_{3n+2}\\mod{3})= (2,0)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=b_n+3d_n$\r\n- $(x_{3n+2}\\mod{3},y_{3n+2}\\mod{3})= (1,1)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®$=2a_n+2c_n$\r\n- $(x_{3n+2}\\mod{3},y_{3n+2}\\mod{3})= (1,1)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=2b_n+2d_n$\r\n\r\nã§ããã\r\n\r\n- $(x_{3n+3}\\mod{3},y_{3n+3}\\mod{3})= (0,0)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=2a_n+6c_n=b_{n+1}$\r\n- $(x_{3n+3}\\mod{3},y_{3n+3}\\mod{3})= (0,0)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã2åã§ãããã®$=2b_n+6d_n$\r\n- $(x_{3n+3}\\mod{3},y_{3n+3}\\mod{3})= (1,2)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®$=3a_n+5c_n=c_{n+1}$\r\n- $(x_{3n+3}\\mod{3},y_{3n+3}\\mod{3})= (1,2)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=3b_n+5d_n=d_{n+1}$\r\n- $(x_{3n+3}\\mod{3},y_{3n+3}\\mod{3})= (2,1)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã0åã§ãããã®$=3a_n+5c_n=c_{n+1}$\r\n- $(x_{3n+3}\\mod{3},y_{3n+3}\\mod{3})= (2,1)$ãã€$(x\\mod{3},y\\mod{3})=(0,0)$ãªãç¹ãéã£ãåæ°ã1åã§ãããã®$=3b_n+5d_n=d_{n+1}$\r\n\r\n\r\n\r\nãšãªãããã£ãŠã\r\n\r\n$$\\begin{pmatrix}a_n\\\\\\\\b_n\\\\\\\\c_n\\\\\\\\d_n\\\\\\\\\\end{pmatrix}=\\begin{pmatrix}0&0&0&0\\\\\\\\2&0&6&0\\\\\\\\3&0&5&0\\\\\\\\0&3&0&5\\end{pmatrix}^{n}\\begin{pmatrix}1\\\\\\\\0\\\\\\\\0\\\\\\\\0\\\\\\\\\\end{pmatrix}$$\r\n\r\nã§ãããæ±ããã¹ãçãã¯$b_{12}+2d_{12}$ãªã®ã§å·Šãã$\\begin{pmatrix}0&1&0&2\\\\\\\\\\end{pmatrix}$ãæããã°OKã§ã\r\n\r\n# èšç®ãããã¯ããã¯\r\n\r\nè¡åã®$1,2,3,6,12$ä¹ãèšç®ããã°OKã§ãã\r\n\r\n$$\\begin{pmatrix}0&0&0&0\\\\\\\\2&0&6&0\\\\\\\\3&0&5&0\\\\\\\\0&3&0&5\\end{pmatrix}^2=\\begin{pmatrix}0&0&0&0\\\\\\\\18&0&30&0\\\\\\\\15&0&25&0\\\\\\\\6&15&18&25\\end{pmatrix}$$\r\n\r\n$$\\begin{pmatrix}0&0&0&0\\\\\\\\2&0&6&0\\\\\\\\3&0&5&0\\\\\\\\0&3&0&5\\end{pmatrix}^3=\\begin{pmatrix}0&0&0&0\\\\\\\\18&0&30&0\\\\\\\\15&0&25&0\\\\\\\\6&15&18&25\\end{pmatrix}\\begin{pmatrix}0&0&0&0\\\\\\\\2&0&6&0\\\\\\\\3&0&5&0\\\\\\\\0&3&0&5\\end{pmatrix}=\\begin{pmatrix}0&0&0&0\\\\\\\\90&0&150&0\\\\\\\\75&0&125&0\\\\\\\\84&75&180&125\\end{pmatrix}$$\r\n\r\n$$\\begin{pmatrix}0&0&0&0\\\\\\\\2&0&6&0\\\\\\\\3&0&5&0\\\\\\\\0&3&0&5\\end{pmatrix}^6=\\begin{pmatrix}0&0&0&0\\\\\\\\90&0&150&0\\\\\\\\75&0&125&0\\\\\\\\84&75&180&125\\end{pmatrix}^2=125 \\begin{pmatrix}0&0&0&0\\\\\\\\90&0&150&0\\\\\\\\75&0&125&0\\\\\\\\246&75&450&125\\end{pmatrix}$$\r\n\r\nããã§ã\r\n$$\\begin{pmatrix}0&0&0&0\\\\\\\\2&0&6&0\\\\\\\\3&0&5&0\\\\\\\\0&3&0&5\\end{pmatrix}^{12}$$\r\n\r\nã«ã€ããŠã¯ã1è¡ç®2åç®ã®æåãš1è¡ç®4åç®ã®æåã ãèšç®ã§ããã°ããã\r\n\r\nãã£ãŠã\r\n\r\n$$125^2\\left(150\\cdot 75+2\\cdot (75\\cdot 90+75\\cdot 450+246\\cdot 125)\\right)=\\bm{2402343750}$$",
"text": "äœè²äŒç³»æ°åŠ",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/11150/515"
}
] | ãã¯ããïŒOMCå㯠$xy$ å¹³é¢ã®åç¹ã«ããïŒä»¥äžã©ã¡ããã®ç§»åãåèš $36$ åç¹°ãè¿ããŸãïŒ
- $x$ 軞ã®æ£ã®æ¹åã« $1$ 移åããïŒ
- $y$ 軞ã®æ£ã®æ¹åã« $1$ 移åããïŒ
ãã㧠$1 \leq n \leq 36$ ãªãæŽæ° $n$ ã«å¯ŸãïŒç§»åã $n$ åç¹°ãè¿ããæç¹ã§OMCåãããç¹ã $P_n$ ãšè¡šããŸãïŒOMCåã移åãç¹°ãè¿ãæ¹æ³ã¯å
šéšã§ $2^{36}$ éããããŸããïŒãã®ãã¡æ¬¡ã®æ¡ä»¶ãã¿ãããã®ã¯å
šéšã§ããã€ãããŸããïŒ
- $P_1, P_2, ..., P_{36}$ ã®ãã¡ã¡ããã© $1$ ã€ã¯ïŒ$x, y$ 座æšããšãã« $3$ ã®åæ°ã§ããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/10277 | D | OMCE003(D) | 400 | 68 | 82 | [
{
"content": "ãæ£æŽæ° $n$ ãšçŽ æ° $p$ ã«å¯ŸããŠïŒ $\\mathrm{ord}_p(n)$ ã $n$ ã $p$ ã§å²ãåããæ倧ã®åæ°ãšå®çŸ©ããïŒ\r\n\r\nã$c$ ãå²ãåããªãä»»æã®çŽ æ° $p$ ã«ã€ããŠïŒ1ã€ç®ã®æ¡ä»¶ããïŒ $$ \\mathrm{ord}_p(a)+\\mathrm{ord}_p(b)\\leq \\mathrm{ord}_p(a^c+b)$$\r\nãšãªãïŒãã㧠$\\mathrm{ord}_p(b)\\lt \\mathrm{ord}_p(a^c)$ ãšä»®å®ãããšïŒ $$ \\mathrm{ord}_p(a)+\\mathrm{ord}_p(b)\\leq \\mathrm{ord}_p(a^c+b)=\\mathrm{ord}_p(b) $$\r\nãã $\\mathrm{ord}_p(a)=0$ ãšãªããïŒããã¯ä»®å®ãšççŸããïŒãŸã $\\mathrm{ord}_p(b)\\gt \\mathrm{ord}_p(a^c)$ ãšä»®å®ãããšïŒ \r\n$$ \\mathrm{ord}_p(a^{c+1})\\lt \\mathrm{ord}_p(a)+\\mathrm{ord}_p(b)\\leq \\mathrm{ord}_p(a^c+b)=\\mathrm{ord}_p(a^c) $$ \r\nãšãªããã¯ãççŸããïŒãããã£ãŠ $\\mathrm{ord}_p(b)=\\mathrm{ord}_p(a^c)$ãåŸãïŒãã®åŒã¯ $c$ ãå²ãåããªãä»»æã®çŽ æ° $p$ ã§æãç«ã€ããïŒ\r\n$$d=(cãšäºãã«çŽ ãªaã®çŽæ°ã®ãã¡æ倧ã®ãã®), \\quad e=\\dfrac{a}{d} $$\r\nãšãããšïŒ $b$ 㯠$d^c$ ãå²ãåãããšãåããïŒ $f=\\dfrac{b}{d^c}$ ãšãããšïŒåé¡æã®åæ¡ä»¶ã¯ä»¥äžãšåå€ã§ããïŒ\r\n\r\n- $def$ 㯠$c(e^c+f)$ ãå²ãåãïŒ\r\n- $d\\geq 2$ ïŒ $a$ 㯠$c$ ã§å²ãåããªãçŽ å æ°ãæã£ãŠããããšããïŒ\r\n- $de,fd^c,c$ 㯠$1000$ 以äžã®æ£æŽæ°ïŒ\r\n\r\nããŸïŒ$2^c\\leq fd^c=b\\leq 1000$ ãã $c\\leq 9$ ã§ããïŒ$c=9$ ã®ãšã $fd^9\\leq 1000$ ãã $d=2,f=1$ ãšãªãïŒãã®ãšãïŒïŒã€ç®ã®æ¡ä»¶ãã $2e$ 㯠$9(e^9+1)$ ãå²ãåãããïŒ $e$ 㯠$1,3,9$ ã®ããããïŒããããã®å ŽåïŒ $(a,b,c)=(2,512,9),(6,512,9),(18,512,9)$ ã§ããïŒããã¯åæ¡ä»¶ãå
šãŠæºããïŒ\\\r\nã以äžãã $c_\\mathrm{max} = 9$ ã§ããïŒæ±ããã¹ãå€ã¯ $\\mathbf{119808}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/10277"
}
] | ã以äžã®æ¡ä»¶ãå
šãŠæºããæ£æŽæ°ã®çµ $(a,b,c)$ ã**ãŸã¶ããçµ**ãšãã³ãŸãïŒ
- $ab$ 㯠$c(a^c+b)$ ãå²ãåãïŒ
- $a$ 㯠$c^{10}$ ãå²ãåããªãïŒ
- $a,b,c$ ã¯å
šãŠ $1000$ 以äžã®æ£æŽæ°ïŒ
ãŸã¶ããçµ $(a, b, c)$ ã«ããã $c$ ãšããŠããåŸãæ倧å€ã $c_\mathrm{max}$ ãšããŸãïŒ $c=c_\mathrm{max}$ ãæºãããŸã¶ããçµ $(a,b,c)$ å
šãŠã«ã€ã㊠$abc$ ã®ç·åãæ±ããŠãã ããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/10300 | E | OMCE003(E) | 500 | 24 | 43 | [
{
"content": "ã$n = 10000$ ãšãïŒ$p_k = \\displaystyle\\sum_{i=1}^n a_i^k$ ãšããïŒãŸã $a_1, a_2, \\ldots, a_n$ ã® $d$ 次ã®åºæ¬å¯Ÿç§°åŒã $\\sigma_d$ ãšããïŒãã®ãšã $1 \\leq k \\leq n$ ã«ã€ã㊠$p_k$ 㯠$\\sigma_1, \\ldots, \\sigma_k$ ã®å€é
åŒã§è¡šãããããšã«æ³šæããïŒ\r\n\r\nã$p_k = 2^k + 3^k $ $(k = 1, 2, \\ldots, n-1)$ ããã³ $\\sigma_n = 2024$ ãã¿ããå€é
åŒ\r\n$$ P(x) = (x-a_1) (x-a_2) \\cdots (x - a_n) $$\r\nã決å®ããããšãèããïŒ$p_1, \\ldots, p_{n-1}$ 㯠$\\sigma_1, \\ldots, \\sigma_{n-1}$ ã«ãã£ãŠè¡šããïŒ$\\sigma_n$ ã«ã¯ãããªãã®ã§ïŒ$C$ ãå®æ°ãšããŠ\r\n$$ P(x) = (x-2)(x-3)x^{n-2} + C $$\r\n㯠$p_k = 2^k + 3^k$ ã®æ¡ä»¶ããã¹ãŠã¿ãããŠããïŒ$C = 0$ ã®å Žåãèããã° $a_1 = 2, a_2 = 3, a_3 = a_4 = \\cdots = 0$ ãšã§ããããïŒïŒããã« $C = \\sigma_n = 2024$ ã代å
¥ããŠïŒ\r\n$$ P(x) = (x-2)(x-3)x^{n-2} + 2024 $$\r\nãæ±ããå€é
åŒã§ããïŒãããšïŒ\r\n$$ \\begin{aligned}\r\np_n &= \\sum_{i=1}^n \\left( a_i^n - P(a_i) \\right) \\\\\\\\\r\n&= \\sum_{i=1}^n \\left( 5a_i^{n-1} - 6 a_i^{n-2} - 2024 \\right) \\\\\\\\\r\n&= 5p_{n-1} - 6p_{n-2} - 2024n \\\\\\\\\r\n&= 2^n + 3^n - 2024n\r\n\\end{aligned} $$\r\nã§ããïŒãããçŽ æ° $p = 4999$ ã§å²ã£ãããŸãã¯ïŒ\r\n$$ 2^{2p+2} + 3^{2p+2} - 2024(2p+2) \\equiv 2^4 + 3^4 - 4048 \\equiv \\mathbf{1048} \\pmod{p} $$\r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/10300"
}
] | ãè€çŽ æ° $a_1,a_2,âŠ,a_{10000}$ ã¯ä»¥äžã®æ¡ä»¶ããšãã«æºãããŸãïŒ
- $k=1,2,âŠ,9999$ ããããã«å¯ŸããŠïŒ$\displaystyle \sum_{i=1}^{10000}a_i^k=2^k+3^k$ïŒ
- $\displaystyle \prod_{i=1}^{10000}a_i=2024$ïŒ
ããã®ãšãïŒ $\displaystyle \sum_{i=1}^{10000}a_i^{10000}$ ã®å€ã¯äžæã«å®ãŸãïŒæ£æŽæ°å€ã«ãªããŸãïŒãã®å€ãçŽ æ° $4999$ ã§å²ã£ãäœããæ±ããŠãã ããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/9224 | F | OMCE003(F) | 700 | 12 | 21 | [
{
"content": "ãæ£æŽæ° $n$ ã«å¯Ÿã $n$ 以äžã®æ£æŽæ°å
šäœãããªãéåã $[n]$ ãšæžãïŒ$X, Y, Z$ ãæ£æŽæ°ãšãïŒäžè¬ã« $\\Gamma$ ã次ã®ããã«å®ããïŒåé¡ã®èšå®ã§ã¯ $(X, Y, Z) = (7, 11, 13)$ ã§ããïŒïŒ\r\n\r\n- $\\Gamma$ ã $x \\in [3X], y \\in [3Y], z \\in [3Z]$ ãªãæ Œåç¹ $(x, y, z)$ ãããªãéåãšããïŒ\r\n\r\nãŸãïŒ$\\Omega$ ã®éšåéå $\\Omega_x, \\Omega_y, \\Omega_z$ ããããã次ã®ããã«å®ããïŒãªãïŒããã㯠$\\Omega_x \\cup \\Omega_y \\cup \\Omega_z = \\Omega$ ãã¿ãããŠããïŒ\r\n- $y, z$ 座æšããšãã« $3$ ã§å²ããš $2$ äœããããªæ Œåç¹å
šäœãããªãéåã $\\Omega_x$ ãšããïŒ\r\n- $z, x$ 座æšããšãã« $3$ ã§å²ããš $2$ äœããããªæ Œåç¹å
šäœãããªãéåã $\\Omega_y$ ãšããïŒ\r\n- $x, y$ 座æšããšãã« $3$ ã§å²ããš $2$ äœããããªæ Œåç¹å
šäœãããªãéåã $\\Omega_z$ ãšããïŒ\r\n\r\nããã§æ Œåç¹ã®éå $U$ ãšæéåã®æ Œåç¹ã®éå $V$ ã«å¯ŸãïŒ$V$ ã次ã®æ¡ä»¶ãã¿ããããšãïŒã$V$ ãæ¡ä»¶ $\\Lambda(U)$ ãã¿ããããšåŒã¶ããšã«ããïŒ\r\n- $V$ ã«å±ãããã¹ãŠã®ç¹ã $P_1, P_2, ..., P_t$ ãšãããšãïŒä»»æã® $Q \\in U$ ã«å¯Ÿãç·å $P_1Q, P_2Q, ..., P_tQ$ ã®äžã§é·ãã $2$ æªæºã®ãã®ã¯é«ã
$1$ åã§ããïŒ\r\n\r\nããã§è£é¡ãäžããïŒ\r\n\r\n---\r\n\r\n**è£é¡ 1.**ã$P, Q$ ãçžç°ãªãæ Œåç¹ãšãããšãïŒç·å $PQ$ ã®é·ãã $2$ æªæºã§ããããã®å¿
èŠååæ¡ä»¶ã¯ïŒ$P, Q$ ã®åº§æšããããã $(x_1, y_1, z_1), (x_2, y_2, z_2)$ ãšãããšãã« $|x_1 - x_2|, |y_1 - y_2|, |z_1 - z_2|$ ããã¹ãŠ $1$ 以äžãšãªãããšã§ããïŒ\r\n\r\n<details><summary>è£é¡ 1. ã®èšŒæ<\\/summary>\r\nã$|x_1 - x_2|, |y_1 - y_2|, |z_1 - z_2|$ ããããã $1$ 以äžã®ãšãã¯\r\n$$PQ = \\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2} \\leq \\sqrt{3} \\lt 2$$\r\nãåŸããïŒéã« $2$ 以äžã®ãã®ãååšããã°\r\n$$PQ \\geq \\max \\\\{ |x_1 - x_2|, |y_1 - y_2|, |z_1 - z_2| \\\\} \\geq 2$$\r\nãåŸãããïŒ\r\n<\\/details>\r\n\r\n---\r\n\r\nãããã§ã¯ïŒ$\\Gamma$ ã®éšåéå $\\gamma$ ã§ãã£ãŠæ¡ä»¶ $\\Lambda(\\Omega)$ ãã¿ãããã®ãèãããïŒ$i \\in [X], j \\in [Y], k \\in [Z]$ ã«å¯ŸãïŒéå $B_{i, j, k}$ ã次ã®ããã«å®ããïŒ\r\n- $3i - 2 \\leq x \\leq 3i, 3j - 2 \\leq y \\leq 3j, 3k - 2 \\leq z \\leq 3k$ ãªãæ Œåç¹ $(x, y, z)$ å
šäœã®éåã $B_{i, j, k}$ ãšããïŒ\r\n\r\nãããš $B_{i, j, k}$ ã¯ç¹ $(3i - 1, 3j - 1, 3k - 1)$ ããã®è·é¢ã $2$ æªæºã®æ Œåç¹å
šäœã®éåã«çããããšãïŒè£é¡ 1. ãããããïŒãŸã $XYZ$ åã®éå $B_{i, j, k}$ 㯠$\\Gamma$ ãåå²ããïŒããªãã¡ã©ã® $\\Gamma$ äžã®ç¹ãïŒãã äžã€ã® $B_{i, j, k}$ ã«å±ããïŒïŒããã«ïŒ$(3i - 1, 3j - 1, 3k - 1)$ 㯠$\\Omega$ äžã®ç¹ã§ããããïŒæ¡ä»¶ $\\Lambda(\\Omega)$ ããã©ã® $B_{i, j, k}$ ã«ã€ããŠãããã«å±ãã $\\gamma$ äžã®ç¹ã¯é«ã
$1$ åã§ããïŒ\\\r\nããã㧠$1000 = 7 \\times 11 \\times 13 - 1$ ã§ããããšã«èæ
®ãïŒä»¥åŸã¯ $\\gamma$ ã«å«ãŸããç¹ã®åæ°ã $XYZ - 1$ ã§ãããšãïŒãã®äžã§æ¡ä»¶ $\\Lambda(\\Omega)$ ãã¿ãã $\\gamma$ ã®åæ°ã $N$ ãšãããïŒãããš $\\gamma$ ãšäº€ããããããªã $B_{i, j, k}$ ããã $1$ ã€ååšããã®ã§ïŒããã $B_{I, J, K}\\ (I \\in [X], J \\in [Y], K \\in [Z])$ ãšããïŒ$(i, j, k) \\neq (I, J, K)$ ãªã $i, j, k$ ã«å¯Ÿã $B_{i, j, k} \\cap \\gamma$ ã«å±ããå¯äžã®ç¹ã $P_{i, j, k}$ ãšããïŒæ¬¡ã®è£é¡ãäžããïŒ\r\n\r\n---\r\n\r\n**è£é¡ 2.**ã$\\gamma$ ãæ¡ä»¶ $\\Lambda(\\Omega_x)$ ãã¿ããããã®å¿
èŠååæ¡ä»¶ã¯ïŒ$i_1, i_2 \\in [X], j \\in [Y], k \\in [Z], i_1 \\neq i_2$ ãªãçµ $(i_1, i_2, j, k)$ ãã©ã®ããã«éžãã§ãïŒ$P_{i_1, j, k}$ ãš $P_{i_2, j, k}$ ã® $x$ 座æšã®å·®ã $3$ 以äžã§ããããšã§ããïŒ\r\n\r\n<details><summary>è£é¡ 2. ã®èšŒæ<\\/summary>\r\nãä»¥äž $2$ ã€ã®åå€æ§ã瀺ãã°ããïŒ\r\n\r\n- **P1ïŒ**$\\gamma$ ãæ¡ä»¶ $\\Lambda(\\Omega_x)$ ãã¿ãããªãïŒ\r\n- **P2ïŒ**$i_1, i_2 \\in [X], j \\in [Y], k \\in [Z], i_1 \\neq i_2$ ãªãçµ $(i_1, i_2, j, k)$ ã§ãã£ãŠïŒ$P_{i_1, j, k}$ ãš $P_{i_2, j, k}$ ã® $x$ 座æšã®å·®ã $2$ 以äžã«ãªããã®ãååšããïŒ\r\n\r\n---\r\n\r\n**P1 $\\Rightarrow$ P2 ã®èšŒæïŒ**\\\r\nã$a, b, c$ ãæŽæ°ãšãïŒ$Q (a, 3b - 1, 3c - 1) \\in \\Omega_x$ ãšçžç°ãªã $P_{i_1, j_1, k_1} (x_1, y_1, z_1), P_{i_2, j_2, k_2} (x_2, y_2, z_2) \\in \\gamma$ ã\r\n$$P_{i_1, j_1, k_1}Q \\lt 2ïŒP_{i_2, j_2, k_2}Q \\lt 2$$\r\nããšãã«ã¿ãããšããïŒæ¬¡ã® $3$ ã€ã瀺ãã°ããïŒ\r\n$$j_1 = j_2ïŒk_1 = k_2ïŒ|x_1 - x_2| \\leq 2$$\r\nãªãïŒ$i_1 \\neq i_2$ ã瀺ãå¿
èŠããããïŒãã㯠$P_{i_1, j_1, k_1}, P_{i_2, j_2, k_2}$ ãçžç°ãªãããšããäžèšã®ãã¡å·Š $2$ ã€ã瀺ãããã°ãã ã¡ã«ããããïŒ\\\r\nã$l \\in \\\\{1, 2\\\\}$ ãä»»æã«å®ããïŒ$P_{i_l, j_l, k_l} \\in B_{i_l, j_l, k_l}$ ã§ããããšïŒããã³ $P_{i_l, j_l, k_l}Q \\lt 2$ ãšè£é¡ 1. ãã\r\n$$3j_l - 2 \\leq y_l \\leq 3j_lïŒ3b - 2 \\leq y_l \\leq 3b$$\r\nãåŸãïŒãã£ãŠ\r\n$$b - \\frac{2}{3} \\leq j_l \\leq b + \\frac{2}{3}$$\r\nãæãç«ã¡ïŒ$b, j_l$ ãæŽæ°ã§ããããšãã $j_l = b$ ãåŸãïŒãããã $j_1 = j_2$ ã§ããïŒåæ§ã®è°è«ã§ $k_1 = k_2$ ãåŸãïŒãŸãïŒäžè§äžçåŒãšè£é¡ 1. ãçšããããšã§\r\n$$|x_1 - x_2| \\leq |x_1 - a| + |x_2 - a| \\leq 2$$\r\nãå°ãããïŒ\r\n\r\n**P2 $\\Rightarrow$ P1 ã®èšŒæïŒ**\\\r\nã$P_{i_1, j, k} (x_1, y_1, z_1)$ ãš $P_{i_2, j, k} (x_2, y_2, z_2)$ ã® $x$ 座æšã®å·®ã $2$ 以äžã§ãããšããïŒããã§äžè¬æ§ã倱ãããšãªã $x_1 \\gt x_2$ ãä»®å®ããïŒãã㧠$\\Omega_x$ äžã®ç¹ $Q ( \\lfloor (x_1 + x_2) \\/ 2 \\rfloor, 3j - 1, 3k -1 )$ ããšãïŒå $l \\in \\\\{1, 2\\\\}$ ã«å¯Ÿã $P_{i_l, j_l, k_l} \\in B_{i_l, j_l, k_l}$ ã§ããããšããïŒ\r\n$$|y_l - (3j - 1)| \\leq 1ïŒ|z_l - (3k - 1)| \\leq 1$$\r\nãããããïŒããã«\r\n$$x_1 \\gt \\left \\lfloor \\frac{x_1 + x_2}{2} \\right \\rfloor \\gt x_2$$\r\nã§ããïŒ\r\n$$\r\n\\begin{aligned}\r\nx_1 - \\left \\lfloor \\frac{x_1 + x_2}{2} \\right \\rfloor &\\lt x_1 - \\frac{x_1 + x_2}{2} + 1 \\\\\\\\\r\n&= \\frac{x_1 - x_2}{2} + 1 \\leq 2 \\\\\\\\\r\n\\left \\lfloor \\frac{x_1 + x_2}{2} \\right \\rfloor - x_2 &\\leq \\frac{x_1 + x_2}{2} - x_2 \\\\\\\\\r\n&= \\frac{x_1 - x_2}{2} \\leq 1\r\n\\end{aligned}\r\n$$\r\nãªã®ã§ $| x_l - \\lfloor (x_1 + x_2) \\/ 2 \\rfloor | \\leq 1$ ãåŸãïŒãã£ãŠïŒè£é¡ 1. ãã $P_{i_1, j, k} Q, P_{i_2, j, k} Q$ ã®é·ãã¯ãšãã« $2$ æªæºã§ããïŒ\r\n<\\/details>\r\n\r\n---\r\n\r\nã$I, J, K$ ãåºå®ããäžã§ïŒè£é¡ 2. ã«åºã¥ã $\\Lambda(\\Omega_x)$ ãã¿ãã $\\gamma$ ã®åæ°ãæ±ãããïŒãŸãïŒ$(j, k) \\neq (J, K)$ ãªãçµ $(j, k)$ ããããã«å¯Ÿã $P_{1, j, k}, P_{2, j, k}, ..., P_{X, j, k}$ ã® $x$ 座æšã®å®ãæ¹ã¯ ${}\\_{X + 2}\\mathrm{C}\\_{2}$ éããã€ããããšããããïŒãã㧠$m \\geq n \\geq 1$ ãªãæŽæ° $m, n$ ã«å¯Ÿãä»¥äž $3$ æ¡ä»¶ãã¿ããããã«èµ€çã»éçã»çœçãå·Šå³äžåã«äžŠã¹ãæ¹æ³ã $f(m, n)$ ãšãã ( $f(m,n) = {}\\_{n+1}\\mathrm{C}\\_{2} \\cdot {}\\_{m-n+2}\\mathrm{C}\\_{2}$ ãšæžãäžãããïŒããã§ã¯ããã¯çšããªã)ïŒ\r\n- èµ€çã»éçã»çœçã®åæ°ã¯ãããã $4$ åïŒ$1$ åïŒ$m - 1$ åã§ããïŒ\r\n- éçã®å³åŽãšå·ŠåŽã«ãããã $2$ åãã€èµ€çãé
眮ãããŠããïŒ\r\n- éçã®å·ŠåŽã«ã¡ããã© $n - 1$ åã®çœçãé
眮ãããŠããïŒ\r\n\r\nãããš $P_{1, J, K}, ..., P_{I - 1, J, K}$ ããã³ $P_{I + 1, J, K}, ..., P_{X, J, K}$ ã® $x$ 座æšã®å®ãæ¹ã®åæ°ã¯ $f(X, I)$ ã§ããïŒãã㯠$P_{1, J, K}, ..., P_{I - 1, J, K}$ ãš $P_{I + 1, J, K}, ..., P_{X, J, K}$ ãããããéçã®å·ŠåŽã»å³åŽã«ããçœçãšçœ®ãæããŠèãïŒãããã® $x$ 座æšã®å²ãæ¯ãæ¹ã¯èµ€çã®é
眮ã«å¯Ÿå¿ãããã®ãšèããã°ããïŒãã£ãŠå
šäœã® $x$ 座æšã®å®ãæ¹ã¯å
šéšã§\r\n$$({}\\_{X + 2}\\mathrm{C}\\_{2})^{YZ - 1} f(X, I)$$\r\nåããïŒ\\\r\nãè£é¡ 2. ãšåæ§ã«ïŒ$\\gamma$ ã $\\Lambda(\\Omega_y), \\Lambda(\\Omega_z)$ ãã¿ããããšãšåå€ãªæ¡ä»¶ãšããŠïŒãããã $\\gamma$ ã«å±ãããã¹ãŠã®ç¹ã® $y, z$ 座æšã«èª²ããããæ¡ä»¶ãšèšãæããããšãã§ããïŒãããã£ãŠãããã®æ¡ä»¶ãã¿ãã $y, z$ 座æšã®å®ãæ¹ã¯ãããã\r\n$$({}\\_{Y + 2}\\mathrm{C}\\_{2})^{ZX - 1} f(Y, J)ïŒ({}\\_{Z + 2}\\mathrm{C}\\_{2})^{XY - 1} f(Z, K)$$\r\nåãã€ããããšããããïŒ$\\gamma$ ã $\\Lambda(\\Omega)$ ãã¿ããããšã¯ããã $\\Lambda(\\Omega_x), \\Lambda(\\Omega_y), \\Lambda(\\Omega_z)$ ããã¹ãŠã¿ããããšãšåå€ã§ããïŒ$x, y, z$ 座æšã¯äºãã«ç¬ç«ããŠå®ããããã®ã§ïŒ$\\Lambda(\\Omega)$ ãã¿ãã $\\gamma$ ã¯\r\n$$({}\\_{X + 2}\\mathrm{C}\\_{2})^{YZ - 1}({}\\_{Y + 2}\\mathrm{C}\\_{2})^{ZX - 1}({}\\_{Z + 2}\\mathrm{C}\\_{2})^{XY - 1} f(X, I)f(Y, J)f(Z, K)$$\r\nåããïŒãããçµ $(I, J, K)$ ã®ãšãåŸãç¯å²ãã¹ãŠã«é¢ããŠç·åããšã£ããã®ã $N$ ã«çããïŒããã§æ£æŽæ° $m$ ã«å¯Ÿã\r\n$$F(m) = \\sum_{n = 1}^{m} f(m, n)$$\r\nãšå®çŸ©ãããšïŒ\r\n$$N = ({}\\_{X + 2}\\mathrm{C}\\_{2})^{YZ - 1}({}\\_{Y + 2}\\mathrm{C}\\_{2})^{ZX - 1}({}\\_{Z + 2}\\mathrm{C}\\_{2})^{XY - 1} F(X)F(Y)F(Z)$$\r\nãšè¡šããïŒãšãã㧠$F(m)$ ã¯ïŒå
ã»ã©ã® $f(m, n)$ ã®å®çŸ©ã«ãããéçã®å·ŠåŽã«ããçœçã®åæ°ã®å¶çŽããªãããŠèµ€çã»éçã»çœçãå·Šå³äžåã«äžŠã³å€ããæ¹æ³ã®åæ°ã«çããã®ã§ $F(m) = {}\\_{m + 4}\\mathrm{C}\\_{5}$ ã§ããïŒãã£ãŠ\r\n$$N = ({}\\_{X + 2}\\mathrm{C}\\_{2})^{YZ - 1}({}\\_{Y + 2}\\mathrm{C}\\_{2})^{ZX - 1}({}\\_{Z + 2}\\mathrm{C}\\_{2})^{XY - 1} {}\\_{X + 4}\\mathrm{C}\\_{5} \\cdot {}\\_{Y + 4}\\mathrm{C}\\_{5} \\cdot {}\\_{Z + 4}\\mathrm{C}\\_{5}$$\r\nã§ããïŒç¹ã« $(X, Y, Z) = (7, 11, 13)$ ã®å Žåã¯\r\n$$N = 2^{377} \\cdot 3^{452} \\cdot 5^{76} \\cdot 7^{79} \\cdot 11^{2} \\cdot 13^{92} \\cdot 17$$\r\nãªã®ã§ïŒè§£çãã¹ãå€ã¯ $\\mathbf{1079}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omce003/editorial/9224"
}
] | ãäžæ¬¡å
空éã«ãããŠïŒ$x, y, z$ 座æšããã¹ãŠæŽæ°ã§ãããããªç¹ã**æ Œåç¹**ãšåŒã¶ããšã«ããŸãïŒæ¬¡ã®æ¡ä»¶ãã¿ããæ Œåç¹å
šäœãããªãéåã $\Omega$ ãšããŸãïŒ
- $x, y, z$ 座æšã®ãã¡**å°ãªããšãäºã€**ã¯ïŒ$3$ ã§å²ããš $2$ äœãæ°ã§ããïŒ
ãŸãïŒä»¥äžã® $3$ æ¡ä»¶ããã¹ãŠã¿ããæ Œåç¹å
šäœãããªãéåã $\Gamma$ ãšããŸãïŒ
- $x$ 座æšã¯ $1$ 以äžã〠$21$ 以äžã§ããïŒ
- $y$ 座æšã¯ $1$ 以äžã〠$33$ 以äžã§ããïŒ
- $z$ 座æšã¯ $1$ 以äžã〠$39$ 以äžã§ããïŒ
ããã§ïŒ$\Gamma$ ããçžç°ãªã $1000$ åã®ç¹ãéžã¶æ¹æ³ã§ãã£ãŠïŒæ¬¡ã®æ¡ä»¶ãã¿ãããã®ã®åæ°ã $N$ ãšããŸãïŒ
- éžãã $1000$ ç¹ã $P_1, P_2, \ldots, P_{1000}$ ãšãããšãïŒä»»æã® $Q \in \Omega$ ã«å¯Ÿã $1000$ åã®ç·å $P_1Q, P_2Q, \ldots, P_{1000}Q$ ã®äžã§é·ãã $2$ æªæºã®ãã®ã¯é«ã
$1$ åã§ããïŒ
ãã ãç¹ãéžã¶é åºã¯åºå¥ããŸããïŒãŸã䞡端ç¹ãäžèŽããç·åã®é·ã㯠$0$ ã§ãããšããŸãïŒ\
ããã®ãšã $N$ ã¯ïŒ$p_1 \lt p_2 \lt \cdots \lt p_t$ ãªã $t$ åã®çŽ æ° $p_1, p_2, \ldots, p_t$ ãš $t$ åã®æ£æŽæ° $n_1, n_2, \ldots, n_t$ ã«ãã£ãŠ
$$N = p_1^{n_1} \times p_2^{n_2} \times \cdots \times p_t^{n_t}$$
ãšïŒäžæçã«ïŒè¡šããã®ã§ïŒ$n_1 + n_2 + \cdots + n_t$ ã®å€ã解çããŠäžããïŒ
<details><summary>解ç圢åŒã«ã€ããŠ<\/summary>
ãããšãã°ïŒããæ±ããå€ã $N = 2^{34} \times 5^{67} \times 89$ ãšãªã£ããšãã¯ïŒ
$$34 + 67 + 1 = 102$$
ãªã®ã§ïŒãã®å Žåã®è§£çãã¹ãå€ã¯ $102$ ãšãªããŸãïŒ
<\/details> |
OMCB010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb010/tasks/4981 | A | OMCB010(A) | 100 | 331 | 346 | [
{
"content": "ã$A\\text{ - }B,B\\text{ - }C,...,E\\text{ - }F$ ã® $5$ ç®æã§é
ç¹ã $100$ ç¹äžããããïŒå®éã«é
ç¹ãäžããã®ã¯ $3$ ç®æãªã®ã§æ¡ä»¶ãæºããé
ç¹ã®çµã¿åãã㯠${}\\_{5}\\mathrm{C}\\_{3}=\\bf10$ éãã§ããïŒ\r\n\r\n\r\n----\r\n**å¥è§£.**\r\nã以äžã® $2$ éãã®å ŽåãããåŸãïŒ\r\n- $100, 200, 300, 400$ ã®ãããã $2$ ã€ã $2$ åïŒæ®ãã® $2$ ã€ã $1$ åã®å ŽåïŒ\r\n- $100, 200, 300, 400$ ã®ãããã $1$ ã€ã $3$ åïŒæ®ãã® $3$ ã€ã $1$ åã®å ŽåïŒ\r\n\r\nããããã«ã€ããŠïŒæ¡ä»¶ãæºããé
ç¹ã®çµã¿åãã㯠${}\\_{4}\\mathrm{C}\\_{2}$ éãïŒ${}\\_{4}\\mathrm{C}\\_{1}$ éãååšããã®ã§ïŒæ¡ä»¶ãæºããçµã¿åãã㯠$6+4=\\textbf{10}$ éãååšããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb010/editorial/4981"
}
] | ãããOMCã®ã³ã³ãã¹ãã«ã€ããŠïŒä»¥äžã®æ¡ä»¶ããšãã«æç«ããŸããïŒ
- ã³ã³ãã¹ãåé¡ã¯é ã« $A, B, C, D, E, F$ ã® $6$ åã§ããïŒé
ç¹ãäœãé ã«äžŠãã§ããïŒ
- $6$ åã«ã¯ $100$ ç¹ïŒ$200$ ç¹ïŒ$300$ ç¹ïŒ$400$ ç¹ããããã $1$ å以äžå«ã¿ïŒãŸããã以å€ã®é
ç¹ã®åé¡ã¯å«ãŸãªãïŒ
ãã®ãšãïŒ$6$ åã®é
ç¹ã®çµã¿åãããšããŠãããããã®ã¯ããã€ãããŸããïŒ |
OMCB010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb010/tasks/6404 | B | OMCB010(B) | 200 | 272 | 300 | [
{
"content": "ãåè§åœ¢ $ABCD$ ã¯äžçµã®å¯ŸèŸºãçããå
æ¥åè§åœ¢ãªã®ã§ç¹ã«çèå°åœ¢ã§ããïŒ$A$ ãã 蟺 $BC$ ã«äžãããåç·ã®è¶³ã $H$ ãšããïŒ$ {AD}=2a$ ãšãããšïŒ$BH = \\dfrac{1}{2}(BC - AD) = a$ ã§ããããïŒäžè§åœ¢ $ABC$ ã«å¯Ÿããäžå¹³æ¹ã®å®çããïŒ\r\n$$AH = \\sqrt{AB^2 - BH^2} = \\sqrt{4-a^2}$$\r\nãåŸãïŒåŸã£ãŠïŒåè§åœ¢ $ {ABCD}$ ã®é¢ç©ã«ã€ããŠïŒä»¥äžã®ãããªæ¹çšåŒãç«ãŠãããïŒ\r\n$$6=\\frac12\\times AH\\times(AD+BC) = 3a\\sqrt{4 - a^2}$$\r\nããã解ã㊠$a^2=2$ ãåããã®ã§ïŒ$ {BC}^2=(4a)^2=\\mathbf{32}$ ãåŸãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb010/editorial/6404"
},
{
"content": "$\\mathrm{AD}=x$ãšãããšã$\\mathrm{BC}=2x$ã§ãããããã§ã\r\n\r\n$$s=\\dfrac{x+2x+2+2}{2}=\\dfrac{3x}{2}+2$$\r\n\r\nã§ããã®ã§ããã©ãŒãã°ãã¿ã®å
¬åŒãã\r\n\r\n$$6=\\sqrt{(s-2x)(s-x)(s-2)(s-2)}=\\sqrt{\\left(4-\\dfrac{x^2}{4}\\right)\\cdot \\dfrac{9x^2}{4}}$$\r\n\r\n$$4=\\left(4-\\dfrac{x^2}{4}\\right)\\cdot \\dfrac{x^2}{4}$$\r\n\r\næŽçãããš\r\n\r\n$$\\left(\\dfrac{x^2}{4}-2\\right)^2=0$$\r\n\r\nãã$x^2=8$ãåŸããç¹ã«çãã¯$(2x)^2=4x^2$ã§ããã",
"text": "ãã©ãŒãã°ãã¿ã®å
¬åŒ",
"url": "https://onlinemathcontest.com/contests/omcb010/editorial/6404/509"
}
] | ãåã«å
æ¥ããåžåè§åœ¢ $ {ABCD}$ ã«ã€ããŠïŒ
$$ {AB}= {CD}=2, \quad {BC}=2 {AD}$$
ãæãç«ã¡ãŸããïŒåè§åœ¢ $ {ABCD}$ ã®é¢ç©ã $6$ ã®ãšãïŒ$ {BC}$ ã®é·ãã® $2$ ä¹ã解çããŠãã ããïŒ |
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