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Given that \( a \) is a positive real number and
\[ f(x) = \log_{2} \left( ax + \sqrt{2x^2 + 1} \right) \]
is an odd function, find the solution set for \( f(x) > \frac{3}{2} \). | (\frac{7}{8},+\infty) | 1/8 |
In an obtuse triangle, the longest side is 4, and the shortest side is 2. Could the area of the triangle be greater than $2 \sqrt{3}$? | No | 6/8 |
Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$. | 540^\circ | 1/8 |
The bases of two equilateral triangles with side lengths \(a\) and \(3a\) lie on the same line. The triangles are situated on opposite sides of the line and do not share any common points. The distance between the nearest ends of their bases is \(2a\). Find the distance between the vertices of the triangles that do not lie on the given line. | 2a\sqrt{7} | 3/8 |
A circle is inscribed in a convex quadrilateral \(ABCD\) with its center at point \(O\), and \(AO=OC\). Additionally, \(BC=5\), \(CD=12\), and \(\angle DAB\) is a right angle.
Find the area of the quadrilateral \(ABCD\). | 60 | 1/8 |
The numbers \(a_{1}, a_{2}, \ldots, a_{n}\) are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. What is the largest possible \(n\) for which this is true? | 16 | 1/8 |
In isosceles trapezoid $ABCD$, parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650$, respectively, and $AD=BC=333$. The angle bisectors of $\angle{A}$ and $\angle{D}$ meet at $P$, and the angle bisectors of $\angle{B}$ and $\angle{C}$ meet at $Q$. Find $PQ$.
Diagram
[asy] /* Made by MRENTHUSIASM */ size(300); pair A, B, C, D, A1, B1, C1, D1, P, Q; A = (-250,6*sqrt(731)); B = (250,6*sqrt(731)); C = (325,-6*sqrt(731)); D = (-325,-6*sqrt(731)); A1 = bisectorpoint(B,A,D); B1 = bisectorpoint(A,B,C); C1 = bisectorpoint(B,C,D); D1 = bisectorpoint(A,D,C); P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); draw(anglemark(P,A,B,1000),red); draw(anglemark(D,A,P,1000),red); draw(anglemark(A,B,Q,1000),red); draw(anglemark(Q,B,C,1000),red); draw(anglemark(P,D,A,1000),red); draw(anglemark(C,D,P,1000),red); draw(anglemark(Q,C,D,1000),red); draw(anglemark(B,C,Q,1000),red); add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*dir(C),linewidth(4)); dot("$D$",D,1.5*dir(D),linewidth(4)); dot("$P$",P,1.5*NE,linewidth(4)); dot("$Q$",Q,1.5*NW,linewidth(4)); draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); label("$500$",midpoint(A--B),1.25N); label("$650$",midpoint(C--D),1.25S); label("$333$",midpoint(A--D),1.25W); label("$333$",midpoint(B--C),1.25E); [/asy] ~MRENTHUSIASM ~ihatemath123 | 242 | 3/8 |
Let $I$ be the incenter of $\triangle ABC$ , and $O$ be the excenter corresponding to $B$ . If $|BI|=12$ , $|IO|=18$ , and $|BC|=15$ , then what is $|AB|$ ? | 24 | 1/8 |
In the parallelogram $\mathrm{ABCD}$, points $\mathrm{E}$ and $\mathrm{F}$ lie on $\mathrm{AD}$ and $\mathrm{AB}$ respectively. Given that the area of $S_{A F I E} = 49$, the area of $\triangle B G F = 13$, and the area of $\triangle D E H = 35$, find the area of $S_{G C H I}$. | 97 | 1/8 |
Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $j$. Let $D_{40}$ be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$ | 761 | 1/8 |
Given that \( x, y, z \) are positive real numbers and \( x + y + z = 1 \), if \( \frac{a}{xyz} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} - 2 \), find the range of the real number \( a \). | (0,\frac{7}{27}] | 2/8 |
The line $y - x \sqrt{3} + 3 = 0$ intersects the parabola $2y^2 = 2x + 3$ at points $A$ and $B.$ Let $P = (\sqrt{3},0).$ Find $|AP - BP|.$ | \frac{2}{3} | 3/8 |
Inside an angle of $30^\circ$ with vertex $A$, a point $K$ is chosen such that its distances to the sides of the angle are 1 and 2. Through point $K$, all possible straight lines intersecting the sides of the angle are drawn. Find the minimum area of the triangle cut off from the angle by one of these lines. | 8 | 2/8 |
A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $9$, $16$, and $36$, respectively. Find the area of $\triangle DEF$. | 169 | 7/8 |
How many roots does the equation \(8x(1 - 2x^2)(8x^4 - 8x^2 + 1) = 1\) have on the interval \([0, 1]\)? | 4 | 1/8 |
The numbers \(a\) and \(b\) are such that each of the two quadratic polynomials \(x^2 + ax + b\) and \(x^2 + bx + a\) has two distinct roots, and the product of these polynomials has exactly three distinct roots. Find all possible values for the sum of these three roots. | 0 | 7/8 |
For \(a, b, c > 0\), find the maximum value of the expression
\[
A=\frac{a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)}{a^{3}+b^{3}+c^{3}-2 abc}
\] | 6 | 5/8 |
How many integers between 2 and 100 inclusive cannot be written as \( m \cdot n \), where \( m \) and \( n \) have no common factors and neither \( m \) nor \( n \) is equal to 1? | 35 | 7/8 |
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$. | 142 | 1/8 |
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
$\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$ | \textbf{(B)}\frac{1}{5} | 1/8 |
Find the length of the arc of the curve \(r = a \cos^3 \left(\frac{\varphi}{3}\right)\) where \(a > 0\). | \frac{3\pi}{2} | 1/8 |
Let \( x \in \left(-\frac{3\pi}{4}, \frac{\pi}{4}\right) \), and \( \cos \left(\frac{\pi}{4} - x\right) = -\frac{3}{5} \). Find the value of \( \cos 2x \). | -\frac{24}{25} | 7/8 |
A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$? | \frac{1}{2} | 4/8 |
A cat has found $432_{9}$ methods in which to extend each of her nine lives. How many methods are there in base 10? | 353 | 6/8 |
For any positive integer \( a \), let \( \tau(a) \) be the number of positive divisors of \( a \). Find, with proof, the largest possible value of \( 4 \tau(n) - n \) over all positive integers \( n \). | 12 | 2/8 |
Calculate the value of the expression
$$
\frac{\left(3^{4}+4\right) \cdot\left(7^{4}+4\right) \cdot\left(11^{4}+4\right) \cdot \ldots \cdot\left(2015^{4}+4\right) \cdot\left(2019^{4}+4\right)}{\left(1^{4}+4\right) \cdot\left(5^{4}+4\right) \cdot\left(9^{4}+4\right) \cdot \ldots \cdot\left(2013^{4}+4\right) \cdot\left(2017^{4}+4\right)}
$$ | 4080401 | 7/8 |
In the quadrilateral pyramid $S-ABCD$ with a right trapezoid as its base, where $\angle ABC = 90^\circ$, $SA \perp$ plane $ABCD$, $SA = AB = BC = 1$, and $AD = \frac{1}{2}$, find the tangent of the angle between plane $SCD$ and plane $SBA$. | \frac{\sqrt{2}}{2} | 3/8 |
Given a 3x3 matrix where each row and each column forms an arithmetic sequence, and the middle element $a_{22} = 5$, find the sum of all nine elements. | 45 | 6/8 |
Maruška received a magical pouch from her grandmother, which doubled the amount of gold coins it contained every midnight. On Monday at noon, Maruška placed some gold coins into the empty pouch. On Tuesday and Wednesday, she took out 40 gold coins from the pouch each day and did not add any coins back in. On Thursday, she again took out 40 gold coins, and the pouch was empty.
How many gold coins did Maruška place into the pouch on Monday?
How many gold coins should she have placed into the empty pouch so that she could withdraw 40 gold coins every day, without adding any coins back in, and such that the amount of gold coins in the pouch was the same before each withdrawal? | 40 | 5/8 |
Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.) | 12 | 1/8 |
On the sphere \(\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}=1\right\}\), given \(n (n \geq 2)\) points \(A_{1}, A_{2}, \ldots, A_{n}\), find the maximum possible value of \(\sum_{1 \leq i < j \leq n} \left| A_i A_j \right|^2\). | n^2 | 6/8 |
Let $f(x)$ have a domain of $R$, $f(x+1)$ be an odd function, and $f(x+2)$ be an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, then calculate the value of $f\left(\frac{9}{2}\right)$. | \frac{5}{2} | 6/8 |
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). What is the minimum number of points in \( M \)? | 12 | 1/8 |
A plane flies between two cities. With the tailwind, it takes 5 hours and 30 minutes, and against the wind, it takes 6 hours. Given that the wind speed is 24 kilometers per hour, and assuming the plane's flying speed is $x$ kilometers per hour, then the speed of the plane with the tailwind is kilometers per hour, and the speed of the plane against the wind is kilometers per hour. | 528 | 7/8 |
Find the number of positive integer solutions \((x, y, z, w)\) to the equation \(x + y + z + w = 25\) that satisfy \(x < y\). | 946 | 7/8 |
Let $C$ be the set of all 100-digit numbers consisting of only the digits $1$ and $2$ . Given a number in $C$ , we may transform the number by considering any $10$ consecutive digits $x_0x_1x_2 \dots x_9$ and transform it into $x_5x_6\dots x_9x_0x_1\dots x_4$ . We say that two numbers in $C$ are similar if one of them can be reached from the other by performing finitely many such transformations. Let $D$ be a subset of $C$ such that any two numbers in $D$ are not similar. Determine the maximum possible size of $D$ . | 21^5 | 1/8 |
Let \( x \) and \( y \) be positive integers such that \( \frac{100}{151} < \frac{y}{x} < \frac{200}{251} \). What is the minimum value of \( x \) ? | 3 | 6/8 |
Define
\[A = \frac{1}{1^2} + \frac{1}{5^2} - \frac{1}{7^2} - \frac{1}{11^2} + \frac{1}{13^2} + \frac{1}{17^2} - \dotsb,\]which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3, and
\[B = \frac{1}{3^2} - \frac{1}{9^2} + \frac{1}{15^2} - \frac{1}{21^2} + \frac{1}{27^2} - \frac{1}{33^2} + \dotsb,\]which includes only terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3.
Determine $\frac{A}{B}.$ | 10 | 1/8 |
In the quadrilateral $ABCD$ , we have $\measuredangle BAD = 100^{\circ}$ , $\measuredangle BCD = 130^{\circ}$ , and $AB=AD=1$ centimeter. Find the length of diagonal $AC$ . | 1 | 2/8 |
In triangle $XYZ$, $XY=XZ$ and $W$ is on $XZ$ such that $XW=WY=YZ$. What is the measure of $\angle XYW$? | 36^{\circ} | 3/8 |
A positive number \( a \) is given. It is known that the equation \( x^3 + 1 = ax \) has exactly two positive roots, and the ratio of the larger to the smaller is 2018. Prove that the equation \( x^3 + 1 = ax^2 \), which also has exactly two positive roots, has the same ratio of the larger root to the smaller root, which is also 2018. | 2018 | 4/8 |
Quadrilateral $ABCD$ is cyclic. Line through point $D$ parallel with line $BC$ intersects $CA$ in point $P$ , line $AB$ in point $Q$ and circumcircle of $ABCD$ in point $R$ . Line through point $D$ parallel with line $AB$ intersects $AC$ in point $S$ , line $BC$ in point $T$ and circumcircle of $ABCD$ in point $U$ . If $PQ=QR$ , prove that $ST=TU$ | ST=TU | 2/8 |
In a plane, there are 7 points, with no three points being collinear. If 18 line segments are connected between these 7 points, then at most how many triangles can these segments form? | 23 | 2/8 |
Let the quadratic function \( f(x) = ax^2 + bx + c \) (\(a > 0\)). The equation \( f(x) - x = 0 \) has two roots \( x_1 \) and \( x_2 \) such that \( 0 < x_1 < x_2 < \frac{1}{a} \).
(1) Prove that \( x < f(x) < x_1 \) when \( x \in (0, x_1) \).
(2) Suppose the graph of the function \( f(x) \) is symmetric with respect to the line \( x = x_0 \). Prove that \( x_0 < \frac{x_1}{2} \). | x_0<\frac{x_1}{2} | 6/8 |
Consider a square arrangement of tiles comprising 12 black and 23 white square tiles. A border consisting of an alternating pattern of black and white tiles is added around the square. The border follows the sequence: black, white, black, white, and so on. What is the ratio of black tiles to white tiles in the newly extended pattern?
A) $\frac{25}{37}$
B) $\frac{26}{36}$
C) $\frac{26}{37}$
D) $\frac{27}{37}$ | \frac{26}{37} | 1/8 |
Given the plane vectors $\overrightarrow{a}=(1,0)$ and $\overrightarrow{b}=\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{a}+ \overrightarrow{b}$. | \frac{\pi}{3} | 5/8 |
The area of the triangle formed by the tangent line at point $(1,1)$ on the curve $y=x^3$, the x-axis, and the line $x=2$ is $\frac{4}{3}$. | \frac{8}{3} | 1/8 |
Let $k$ be a real number. Define on the set of reals the operation $x*y$ = $\frac{xy}{x+y+k}$ whenever $x+y$ does not equal $-k$ . Let $x_1<x_2<x_3<x_4$ be the roots of $t^4=27(t^2+t+1)$ .suppose that $[(x_1*x_2)*x_3]*x_4=1$ . Find all possible values of $k$ | 3 | 1/8 |
Reduce one side of a square by $2 \mathrm{cm}$ and increase the other side by $20\%$ to form a rectangle with an area equal to the original square. What is the length of a side of the original square in $\mathrm{cm}$? | 12\, | 1/8 |
Through the right focus of the hyperbola \( x^{2}-\frac{y^{2}}{2}=1 \), draw a line \( l \) that intersects the hyperbola at two points \( A \) and \( B \). If the real number \( \lambda \) such that \( |AB|=\lambda \) corresponds to exactly 3 such lines \( l \), then \( \lambda = \quad \). | 4 | 4/8 |
If $5(\cos a + \cos b) + 4(\cos a \cos b + 1) = 0,$ then find all possible values of
\[\tan \frac{a}{2} \tan \frac{b}{2}.\]Enter all the possible values, separated by commas. | 3,-3 | 1/8 |
In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won? | 11 | 2/8 |
There are four weights of different masses. Katya weighs the weights in pairs. The results are 1800, 1970, 2110, 2330, and 2500 grams. How many grams does the sixth weighing result in? | 2190 | 6/8 |
For each of the $9$ positive integers $n,2n,3n,\dots , 9n$ Alice take the first decimal digit (from the left) and writes it onto a blackboard. She selected $n$ so that among the nine digits on the blackboard there is the least possible number of different digits. What is this number of different digits equals to? | 4 | 1/8 |
There are some stamps with denominations of 0.5 yuan, 0.8 yuan, and 1.2 yuan, with a total value of 60 yuan. The number of 0.8 yuan stamps is four times the number of 0.5 yuan stamps. How many 1.2 yuan stamps are there? | 13 | 7/8 |
Two teams, Team A and Team B, are playing in a basketball finals series that uses a "best of seven" format (the first team to win four games wins the series and the finals end). Based on previous game results, Team A's home and away schedule is arranged as "home, home, away, away, home, away, home". The probability of Team A winning at home is 0.6, and the probability of winning away is 0.5. The results of each game are independent of each other. Calculate the probability that Team A wins the series with a 4:1 score. | 0.18 | 5/8 |
A tower is $45 \mathrm{~m}$ away from the bank of a river. If the width of the river is seen at an angle of $20^{\circ}$ from a height of $18 \mathrm{~m}$ in the tower, how wide is the river? | 16.38 | 1/8 |
Is the number of ordered 10-tuples of positive integers \((a_1, a_2, \ldots, a_{10})\) such that \(\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{10}} = 1\) even or odd? | odd | 2/8 |
Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center? | 6060 | 2/8 |
The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
[asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy] | 790 | 1/8 |
Given that the sum of the binomial coefficients of the first two terms of the expansion of \\({(2x+\frac{1}{\sqrt{x}})}^{n}\\) is \\(10\\).
\\((1)\\) Find the value of \\(y' = 2x\\).
\\((2)\\) Find the constant term in this expansion. | 672 | 7/8 |
If the Cesaro sum of a sequence with 99 terms is 1000, calculate the Cesaro sum of the sequence with 100 terms consisting of the numbers 1 and the first 99 terms of the original sequence. | 991 | 7/8 |
A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$? | \sqrt{2} | 6/8 |
Given that $a > 2b$ ($a, b \in \mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\infty)$. Determine the minimum value of $$\frac{a^2 + 4b^2}{a - 2b}$$. | \sqrt{2} | 6/8 |
If the point $\left(m,n\right)$ in the first quadrant is symmetric with respect to the line $x+y-2=0$ and lies on the line $2x+y+3=0$, calculate the minimum value of $\frac{1}{m}+\frac{8}{n}$. | \frac{25}{9} | 7/8 |
George, Jeff, Brian, and Travis decide to play a game of hot potato. They begin by arranging themselves clockwise in a circle in that order. George and Jeff both start with a hot potato. On his turn, a player gives a hot potato (if he has one) to a randomly chosen player among the other three. If a player has two hot potatoes on his turn, he only passes one. If George goes first, and play proceeds clockwise, what is the probability that Travis has a hot potato after each player takes one turn? | \frac{5}{27} | 1/8 |
In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets $ 1$ point and the loser gets $ 0$ points. If the game is tied, each player gets $ 0.5$ points. Given a positive integer $ m$ , a tournament is said to have property $ P(m)$ if the following holds: among every set $ S$ of $ m$ players, there is one player who won all her games against the other $ m\minus{}1$ players in $ S$ and one player who lost all her games against the other $ m \minus{} 1$ players in $ S$ . For a given integer $ m \ge 4$ , determine the minimum value of $ n$ (as a function of $ m$ ) such that the following holds: in every $ n$ -player round robin chess tournament with property $ P(m)$ , the final scores of the $ n$ players are all distinct. | 2m-3 | 4/8 |
If \( n \) is a natural number and the value of \( n^{3} \) ends exactly with "2016", find the smallest value of \( n \). | 856 | 6/8 |
Find the range of the function \( f(x) = \frac{1}{g\left( \frac{64g(g(\ln x))}{1025} \right)} \), where
\[ g(x) = x^5 + \frac{1}{x^5} \] | [-\frac{32}{1025},0)\cup(0,\frac{32}{1025}] | 4/8 |
$A B C D$ is any quadrilateral, $P$ and $Q$ are the midpoints of the diagonals $B D$ and $A C$ respectively. The parallels through $P$ and $Q$ to the other diagonal intersect at $O$.
If we join $O$ with the four midpoints of the sides $X, Y$, $Z$, and $T$, four quadrilaterals $O X B Y, O Y C Z, O Z D T$, and $OTAX$ are formed.
Prove that the four quadrilaterals have the same area. | 4 | 1/8 |
Find all triples $(a,b,c)$ of distinct positive integers so that there exists a subset $S$ of the positive integers for which for all positive integers $n$ exactly one element of the triple $(an,bn,cn)$ is in $S$ .
Proposed by Carl Schildkraut, MIT | (,b,) | 1/8 |
Let \( n \) and \( k \) be positive integers with \( n > k \). Prove that the greatest common divisor of \( C_{n}^{k}, C_{n+1}^{k}, \cdots, C_{n+k}^{k} \) is 1. | 1 | 6/8 |
Let \(r_1, r_2, r_3\) be the roots of the real-coefficient equation \(x^3 - x^2 + ax - b = 0\), where \(0 < r_i < 1\) for \(i=1,2,3\). Find the maximum possible value of \(7a - 9b\). | 2 | 6/8 |
Let $A$ denote the set of all integers $n$ such that $1 \leq n \leq 10000$, and moreover the sum of the decimal digits of $n$ is 2. Find the sum of the squares of the elements of $A$. | 7294927 | 7/8 |
There are $5$ people participating in a lottery, each drawing a ticket from a box containing $5$ tickets ($3$ of which are winning tickets) without replacement until all $3$ winning tickets have been drawn, ending the activity. The probability that the activity ends exactly after the $4$th person draws is $\_\_\_\_\_\_$. | \frac{3}{10} | 6/8 |
Given a parallelogram $ABCD$ where $\angle B = 111^{\circ}$ and $BC = BD$. Point $H$ is marked on segment $BC$ such that $\angle BHD = 90^{\circ}$. Point $M$ is the midpoint of side $AB$. Find the angle $AMH$. Provide the answer in degrees. | 132 | 1/8 |
Let \( S=\{A_{1}, A_{2}, \cdots, A_{n}\} \), where \( A_{1}, A_{2}, \cdots, A_{n} \) are \( n \) distinct finite sets (\( n \geqslant 2 \)), satisfying that for any \( A_{i}, A_{j} \in S \), \( A_{i} \cup A_{j} \in S \). If \( k=\min_{1 \leqslant i \leqslant n}|A_{i}| \geqslant 2 \), prove that there exists an \( x \in \bigcup_{i=1}^{n} A_{i} \) such that \( x \) belongs to at least \(\frac{n}{k} \) of the sets \( A_{1}, A_{2}, \cdots, A_{n} \) (where \( |X| \) denotes the number of elements in the finite set \( X \)). | \frac{n}{k} | 4/8 |
Next year, 2022, has the property that it may be written using at most two different digits, namely 2 and 0. How many such years will there be between 1 and 9999 inclusive? | 927 | 3/8 |
A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner.
| 504 | 7/8 |
The lengths of the sides of the pentagon \(ABCDE\) are equal to 1. Let points \(P, Q, R,\) and \(S\) be the midpoints of the sides \(AB, BC, CD,\) and \(DE\) respectively, and let points \(K\) and \(L\) be the midpoints of segments \(PR\) and \(QS\) respectively. Find the length of segment \(KL\). | \frac{1}{4} | 6/8 |
Consider a bug starting at vertex $A$ of a cube, where each edge of the cube is 1 meter long. At each vertex, the bug can move along any of the three edges emanating from that vertex, with each edge equally likely to be chosen. Let $p = \frac{n}{6561}$ represent the probability that the bug returns to vertex $A$ after exactly 8 meters of travel. Find the value of $n$. | 1641 | 3/8 |
Using one, two, and three fours, determine which numbers can be expressed using various arithmetic operations. For example, how can 64 be represented using two fours? | \sqrt{(\sqrt{\sqrt{4}})^{4!}}=64 | 1/8 |
The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$ . Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$ , where $m$ , $n$ are relatively prime integers. Find $m + n$ . | 2224 | 6/8 |
Let $M$ be the midpoint of side $AB$ of triangle $ABC$. Let $P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of triangle $BPD$ to that of triangle $ABC$ is denoted by $r$, then | r=\frac{1}{2} | 7/8 |
In the convex polygon \( A_{1}, A_{2}, \ldots, A_{2n} \), let the interior angle at vertex \( A_{i} \) be denoted as \( \alpha_{i} \). Show that
$$
\alpha_{1}+\alpha_{3}+\ldots+\alpha_{2n-1} = \alpha_{2}+\alpha_{4}+\ldots+\alpha_{2n}
$$ | \alpha_{1}+\alpha_{3}+\ldots+\alpha_{2n-1}=\alpha_{2}+\alpha_{4}+\ldots+\alpha_{2n} | 1/8 |
In the number $52674.1892$, calculate the ratio of the value of the place occupied by the digit 6 to the value of the place occupied by the digit 8. | 10,000 | 5/8 |
From the numbers 0, 1, 2, 3, 4, select three different digits to form a three-digit number. What is the sum of the units digit of all these three-digit numbers? | 90 | 3/8 |
Define \(m \otimes n = m \times m - n \times n\). What is the value of \(2 \otimes 4 - 4 \otimes 6 - 6 \otimes 8 - \cdots - 98 \otimes 100\)? | 9972 | 6/8 |
Given a circle with a unit radius centered at O. Point \( A \) is at a distance \( a \) from the center ( \( 0 < a < 1 \) ). Through point \( A \), all possible chords \( MN \) are drawn.
a) Find the length of the shortest chord \( MN \).
b) Find the maximum area of the triangle \( OMN \). | \sqrt{1-^{2}} | 1/8 |
If $f(1) = 3$, $f(2)= 12$, and $f(x) = ax^2 + bx + c$, what is the value of $f(3)$? | 21 | 1/8 |
30 students from 5 grades participated in answering 40 questions. Each student answered at least 1 question. Every two students from the same grade answered the same number of questions, and students from different grades answered a different number of questions. How many students answered only 1 question? | 26 | 6/8 |
Let \( N \) be the smallest positive integer whose digits have a product of 2000. The sum of the digits of \( N \) is | 25 | 7/8 |
On weekdays, a scatterbrained scientist takes the Moscow metro ring line from Taganskaya station to Kievskaya station to get to work, and makes the return trip in the evening. The scientist boards the first available train that arrives. It is known that trains run at approximately equal intervals in both directions. The northern route (via Belorusskaya) takes 17 minutes to travel from Kievskaya to Taganskaya or vice versa, while the southern route (via Paveletskaya) takes 11 minutes.
The scientist, as usual, counts everything. One day, he calculated that, based on years of observations:
- A train traveling counterclockwise arrives at Kievskaya on average 1 minute and 15 seconds after a train traveling clockwise. The same applies to Taganskaya;
- The scientist's average commute from home to work takes 1 minute less than the commute from work to home.
Find the expected value of the interval between trains traveling in one direction. | 3 | 4/8 |
In the sequence
\[..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,...\]
each term is the sum of the two terms to its left. Find $a$.
$\textbf{(A)}\ -3 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 3$ | \textbf{(A)}\-3 | 1/8 |
On the hypotenuse $AB$ of the right triangle $ABC$, a square $ABDE$ is constructed outward. It is known that $AC = 1$ and $BC = 3$.
In what ratio does the angle bisector of angle $C$ divide the side $DE$? | 1:3 | 5/8 |
In a diagram, the grid is composed of 1x1 squares. What is the area of the shaded region if the overall width of the grid is 15 units and its height is 5 units? Some parts are shaded in the following manner: A horizontal stretch from the left edge (6 units wide) that expands 3 units upward from the bottom, and another stretch that begins 6 units from the left and lasts for 9 units horizontally, extending from the 3 units height to the top of the grid. | 36 | 7/8 |
There are 11 integers written on a board (not necessarily distinct). Could it be possible that the product of any five of these integers is greater than the product of the remaining six? | Yes | 3/8 |
Given a large circle with a radius of 11 and small circles with a radius of 1, find the maximum number of small circles that can be tangentially inscribed in the large circle without overlapping. | 31 | 7/8 |
The Absent-Minded Scientist had a sore knee. The doctor prescribed him 10 pills for his knee: take one pill daily. The pills are effective in $90\%$ of cases, and in $2\%$ of cases, there is a side effect—absent-mindedness disappears, if present.
Another doctor prescribed the Scientist pills for absent-mindedness—also one per day for 10 consecutive days. These pills cure absent-mindedness in $80\%$ of cases, but in $5\%$ of cases, there is a side effect—the knee stops hurting.
The bottles with the pills look similar, and when the Scientist went on a ten-day business trip, he took one bottle with him but didn't pay attention to which one. For ten days, he took one pill per day and returned completely healthy. Both the absent-mindedness and the knee pain were gone. Find the probability that the Scientist took pills for absent-mindedness. | 0.69 | 1/8 |
Positive integers $a, b$, and $c$ have the property that $a^{b}, b^{c}$, and $c^{a}$ end in 4, 2, and 9, respectively. Compute the minimum possible value of $a+b+c$. | 17 | 1/8 |
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