problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible. | 95 | 7/8 |
A standard domino game consists of 28 tiles. Each tile is made up of two integers ranging from 0 to 6, inclusive. All possible combinations $(a, b)$, where $a \leq b$, are listed exactly once. Note that tile $(4,2)$ is listed as tile $(2,4)$, because $2 \leq 4$. Excluding the piece $(0,0)$, for each of the other 27 tiles $(a, b)$, where $a \leq b$, we write the fraction $\frac{a}{b}$ on a board.
a) How many distinct values are written in the form of fractions on the board? (Note that the fractions $\frac{1}{2}$ and $\frac{2}{4}$ have the same value and should be counted only once.)
b) What is the sum of the distinct values found in the previous item? | \frac{13}{2} | 7/8 |
In rectangle $ADEH$, points $B$ and $C$ trisect $\overline{AD}$, and points $G$ and $F$ trisect $\overline{HE}$. In addition, $AH=AC=2$. What is the area of quadrilateral $WXYZ$ shown in the figure?
[asy]
unitsize(1cm);
pair A,B,C,D,I,F,G,H,U,Z,Y,X;
A=(0,0);
B=(1,0);
C=(2,0);
D=(3,0);
I=(3,2);
F=(2,2);
G=(1,2);
H=(0,2);
U=(1.5,1.5);
Z=(2,1);
Y=(1.5,0.5);
X=(1,1);
draw(A--D--I--H--cycle,linewidth(0.7));
draw(H--C,linewidth(0.7));
draw(G--D,linewidth(0.7));
draw(I--B,linewidth(0.7));
draw(A--F,linewidth(0.7));
label("$A$",A,SW);
label("$B$",B,S);
label("$C$",C,S);
label("$D$",D,SE);
label("$E$",I,NE);
label("$F$",F,N);
label("$G$",G,N);
label("$H$",H,NW);
label("$W$",U,N);
label("$X$",X,W);
label("$Y$",Y,S);
label("$Z$",Z,E);
[/asy] | \frac{1}{2} | 6/8 |
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c, a \neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.) | 4 | 1/8 |
In a geometric progression with a common ratio of 4, denoted as $\{b_n\}$, where $T_n$ represents the product of the first $n$ terms of $\{b_n\}$, the fractions $\frac{T_{20}}{T_{10}}$, $\frac{T_{30}}{T_{20}}$, and $\frac{T_{40}}{T_{30}}$ form another geometric sequence with a common ratio of $4^{100}$. Analogously, for an arithmetic sequence $\{a_n\}$ with a common difference of 3, if $S_n$ denotes the sum of the first $n$ terms of $\{a_n\}$, then _____ also form an arithmetic sequence, with a common difference of _____. | 300 | 5/8 |
Given that there are 6 balls of each of the four colors: red, blue, yellow, and green, each numbered from 1 to 6, calculate the number of ways to select 3 balls with distinct numbers, such that no two balls have the same color or consecutive numbers. | 96 | 7/8 |
Consider the polynomial with blanks:
\[ T = \_ X^2 + \_ X + \_ \]
Tic and Tac play the following game. In one turn, Tic chooses a real number and Tac places it in one of the 3 blanks. After 3 turns, the game ends. Tic wins if the resulting polynomial has 2 distinct rational roots, and Tac wins otherwise.
Who has a winning strategy? | Tic | 1/8 |
How many positive integers $n$ are there for which both the geometric and harmonic means of $n$ and 2015 are integers? | 5 | 7/8 |
(1) If the real numbers \(x, y, z\) satisfy \(x^{2} + y^{2} + z^{2} = 1\), prove that \(|x-y| + |y-z| + |z-x| \leqslant 2 \sqrt{2}\).
(2) If 2023 real numbers \(x_{1}, x_{2}, \cdots, x_{2023}\) satisfy \(x_{1}^{2} + x_{2}^{2} + \cdots + x_{2023}^{2} = 1\), find the maximum value of \(\left|x_{1}-x_{2}\right| + \left|x_{2}-x_{3}\right| + \cdots + \left|x_{2022}-x_{2023}\right| + \left|x_{2023}-x_{1}\right|\). | 2\sqrt{2022} | 1/8 |
In triangle $\vartriangle ABC$ with orthocenter $H$ , the internal angle bisector of $\angle BAC$ intersects $\overline{BC}$ at $Y$ . Given that $AH = 4$ , $AY = 6$ , and the distance from $Y$ to $\overline{AC}$ is $\sqrt{15}$ , compute $BC$ . | 4\sqrt{35} | 1/8 |
In triangle $ABC$, let vector $\vec{a} = (1, \cos B)$ and vector $\vec{b} = (\sin B, 1)$, and suppose $\vec{a}$ is perpendicular to $\vec{b}$. Find the magnitude of angle $B$. | \frac{3\pi}{4} | 2/8 |
Find the only value of \( x \) in the open interval \((- \pi / 2, 0)\) that satisfies the equation
$$
\frac{\sqrt{3}}{\sin x} + \frac{1}{\cos x} = 4.
$$ | -\frac{4\pi}{9} | 6/8 |
If $\frac{137}{a}=0.1 \dot{2} 3 \dot{4}$, find the value of $a$. | 1110 | 7/8 |
In a right triangle \(ABC\) with \(\angle A = 60^{\circ}\), point \(N\) is marked on the hypotenuse \(AB\), and point \(K\) is the midpoint of segment \(CN\). It is given that \(AK = AC\). The medians of triangle \(BCN\) intersect at point \(M\). Find the angle between lines \(AM\) and \(CN\). | 60 | 6/8 |
A wheel of radius \( r \) is traveling along a road without slipping with angular velocity \( \omega \) (where \( \omega > \sqrt{\frac{g}{r}} \)). A particle is thrown off the rim of the wheel. Show that it can reach a maximum height above the road of \(\frac{(r\omega + \frac{g}{\omega})^2}{2g}\). Assume air resistance is negligible. | \frac{(r\omega+\frac{}{\omega})^2}{2g} | 1/8 |
For how many positive integers $x$ is $\log_{10}(x-40) + \log_{10}(60-x) < 2$ ?
$\textbf{(A) }10\qquad \textbf{(B) }18\qquad \textbf{(C) }19\qquad \textbf{(D) }20\qquad \textbf{(E) }\text{infinitely many}\qquad$ | \textbf{(B)}18 | 1/8 |
If $|x-\log y|=x+\log y$ where $x$ and $\log y$ are real, then
$\textbf{(A) }x=0\qquad \textbf{(B) }y=1\qquad \textbf{(C) }x=0\text{ and }y=1\qquad\\ \textbf{(D) }x(y-1)=0\qquad \textbf{(E) }\text{None of these}$ | \textbf{(D)}x(y-1)=0 | 1/8 |
There is a jar containing $m$ white balls and $n$ black balls $(m > n)$. Balls are drawn one by one without replacement. If at some point during the drawing process, the number of white balls drawn equals the number of black balls drawn, this is called a coincidence. Find the probability of having at least one coincidence. | \frac{2n}{+n} | 4/8 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x - y)) = f(x) f(y) - f(x) + f(y) - xy\]for all $x,$ $y.$ Find the sum of all possible values of $f(1).$ | -1 | 5/8 |
Given a tetrahedron $ABCD$ with an internal point $P$, what is the minimum number of its edges that appear obtuse when viewed from point $P$? | 3 | 4/8 |
On February 23rd, a boy named Zhenya received a chocolate bar of size $3 \times 3$, with a different picture on each piece. Each turn, Zhenya can eat one piece that has no more than three sides in common with other unswallowed pieces. In how many ways can Zhenya eat his chocolate bar? | 290304 | 2/8 |
A quadrilateral \(ABCD\) has an area of 1. From an internal point \(O\), perpendiculars \(OK\), \(OL\), \(OM\), and \(ON\) are dropped to the sides \(AB\), \(BC\), \(CD\), and \(DA\) respectively. It is known that \(AK \geq KB\), \(BL \geq LC\), \(CM \geq MD\), and \(DN \geq NA\). Find the area of quadrilateral \(KLMN\). | \frac{1}{2} | 2/8 |
What is the remainder when 1,234,567,890 is divided by 99? | 72 | 7/8 |
Alina takes the bus to school, which arrives every 15 minutes according to the schedule. The time to the bus stop is always the same for her. If she leaves home at 8:20, she reaches school at 8:57, and if she leaves home at 8:21, she will be late for school. Classes start at 9:00. How many minutes will Alina be late for school if she leaves home at 8:23? | 12 | 3/8 |
Given an isosceles triangle $A B C$ where $A B = A C$ and $\angle A B C = 53^{\circ}$. Point $K$ is such that $C$ is the midpoint of segment $A K$. Point $M$ is chosen so that:
- $B$ and $M$ are on the same side of line $A C$;
- $K M = A B$;
- $\angle M A K$ is the largest possible.
How many degrees is the angle $\angle B A M$? | 44 | 2/8 |
Given that the sum of the binomial coefficients in the expansion of $(5x- \frac{1}{\sqrt{x}})^n$ is 64, determine the constant term in its expansion. | 375 | 2/8 |
Prove that if triangles \( a b c \) and \( a' b' c' \) in the complex plane are similar, then
$$
\frac{b-a}{c-a}=\frac{b'-a'}{c'-a'}
$$ | \frac{b-}{-}=\frac{b'-'}{'-'} | 1/8 |
Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{1}{2} \qquad \textbf{(C)}\ \frac{\sqrt{5}-1}{2} \qquad \textbf{(D)}\ \frac{\sqrt{5}+1}{2} \qquad \textbf{(E)}\ \sqrt{5}$ | \textbf{(A)}\0 | 1/8 |
For n points \[ P_1;P_2;...;P_n \] in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points \[ P_i;P_{i+1} (i=1;2;...n-1) \] is the same color or 2 points with at least one of 2 points are colored white. How many ways acceptable color? | \frac{3^{n+1} + (-1)^{n+1}}{2} | 4/8 |
Given the function
$$
f(x) = 10x^2 + mx + n \quad (m, n \in \mathbb{Z})
$$
which has two distinct real roots in the interval \((1, 3)\). Find the maximum possible value of \(f(1) f(3)\). | 99 | 1/8 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $(E)$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, $M$ and $N$ are the endpoints of its minor axis, and the perimeter of the quadrilateral $MF\_1NF\_2$ is $4$, let line $(l)$ pass through $F\_1$ intersecting $(E)$ at points $A$ and $B$ with $|AB|=\frac{4}{3}$.
1. Find the maximum value of $|AF\_2| \cdot |BF\_2|$.
2. If the slope of line $(l)$ is $45^{\circ}$, find the area of $\triangle ABF\_2$. | \frac{2}{3} | 1/8 |
In recent years, handicrafts made by a certain handicraft village have been very popular overseas. The villagers of the village have established a cooperative for exporting handicrafts. In order to strictly control the quality, the cooperative invites 3 experts to inspect each handicraft made by the villagers. The quality control process is as follows: $(i)$ If all 3 experts consider a handicraft to be of good quality, then the quality of the handicraft is rated as grade $A$; $(ii)$ If only 1 expert considers the quality to be unsatisfactory, then the other 2 experts will conduct a second quality check. If both experts in the second check consider the quality to be good, then the handicraft is rated as grade $B$. If one or both of the experts in the second check consider the quality to be unsatisfactory, then the handicraft is rated as grade $C$; $(iii)$ If 2 or all 3 experts consider the quality to be unsatisfactory, then the handicraft is rated as grade $D$. It is known that the probability of 1 handicraft being considered unsatisfactory by 1 expert in each quality check is $\frac{1}{3}$, and the quality of each handicraft is independent of each other. Find: $(1)$ the probability that 1 handicraft is rated as grade $B$; $(2)$ if handicrafts rated as grade $A$, $B$, and $C$ can be exported with profits of $900$ yuan, $600$ yuan, and $300$ yuan respectively, while grade $D$ cannot be exported and has a profit of $100$ yuan. Find: $①$ the most likely number of handicrafts out of 10 that cannot be exported; $②$ if the profit of 1 handicraft is $X$ yuan, find the distribution and mean of $X$. | \frac{13100}{27} | 7/8 |
Rosencrantz plays \( n \leq 2015 \) games of question, and ends up with a win rate (i.e. \(\#\) of games won / \(\#\) of games played) of \( k \). Guildenstern has also played several games, and has a win rate less than \( k \). He realizes that if, after playing some more games, his win rate becomes higher than \( k \), then there must have been some point in time when Rosencrantz and Guildenstern had the exact same win-rate. Find the product of all possible values of \( k \). | \frac{1}{2015} | 1/8 |
The integers \( a \) and \( b \) have the following property: for every natural number \( n \), the integer \( 2^n a + b \) is a perfect square. Show that \( a \) is zero. | 0 | 4/8 |
What is the largest number of towns that can meet the following criteria: Each pair is directly linked by just one of air, bus, or train. At least one pair is linked by air, at least one pair by bus, and at least one pair by train. No town has an air link, a bus link, and a train link. No three towns, A, B, C, are such that the links between AB, AC, and BC are all air, all bus, or all train. | 4 | 1/8 |
How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.)
*Proposed by Miles Yamner and Andrew Wu*
(Note: wording changed from original to clarify) | 1225 | 1/8 |
Three cones are standing on their bases on a table, touching each other. The radii of their bases are $2r$, $3r$, and $10r$. A truncated cone with the smaller base down is placed on the table, sharing a slant height with each of the other cones. Find $r$ if the radius of the smaller base of the truncated cone is 15. | 29 | 6/8 |
Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous functions such that $\lim_{x\rightarrow \infty} f(x) =\infty$ and $\forall x,y\in \mathbb{R}, |x-y|>\varphi, \exists n<\varphi^{2023}, n\in \mathbb{N}$ such that $$ f^n(x)+f^n(y)=x+y $$ | f(x)=x | 1/8 |
The radius of the circle inscribed in triangle \(ABC\) is 4, with \(AC = BC\). On the line \(AB\), point \(D\) is chosen such that the distances from \(D\) to the lines \(AC\) and \(BC\) are 11 and 3 respectively. Find the cosine of the angle \(DBC\). | \frac{3}{4} | 4/8 |
Arrange 5 people to be on duty from Monday to Friday, with each person on duty for one day and one person arranged for each day. The conditions are: A and B are not on duty on adjacent days, while B and C are on duty on adjacent days. The number of different arrangements is $\boxed{\text{answer}}$. | 36 | 3/8 |
The numbers \( x, y, z \) are such that \( \frac{x + \frac{53}{18} y - \frac{143}{9} z}{z} = \frac{\frac{3}{8} x - \frac{17}{4} y + z}{y} = 1 \). Find \( \frac{y}{z} \). | \frac{352}{305} | 7/8 |
A convex regular polygon with 1997 vertices has been decomposed into triangles using diagonals that do not intersect internally. How many of these triangles are acute? | 1 | 1/8 |
Solve the equation \[-2x^2 = \frac{4x + 2}{x + 2}.\] | -1 | 7/8 |
Given that the angle between the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute, and for any $(x,y)$ that satisfies $|x\overrightarrow{a}+y\overrightarrow{b}|=1$ and $xy\geqslant 0$, the inequality $|x+2y|\leqslant \frac{8}{\sqrt{15}}$ holds. Find the minimum value of $\overrightarrow{a}\cdot\overrightarrow{b}$. | \frac{1}{4} | 6/8 |
On the round necklace there are $n> 3$ beads, each painted in red or blue. If a bead has adjacent beads painted the same color, it can be repainted (from red to blue or from blue to red). For what $n$ for any initial coloring of beads it is possible to make a necklace in which all beads are painted equally? | n | 1/8 |
The graphs of four functions, labelled (2) through (5), are shown below. Note that the domain of function (3) is $$\{-5,-4,-3,-2,-1,0,1,2\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;
picture pic1,pic2,pic3,pic4;
draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));
real f(real x) {return x^2-2x;}
real h(real x) {return -atan(x);}
real k(real x) {return 4/x;}
real x;
draw(pic1,graph(f,-2,4),Arrows(4));
draw(pic3,graph(h,-8,8),Arrows(4));
draw(pic4,graph(k,-8,-0.125*4),Arrows(4));
draw(pic4,graph(k,0.125*4,8),Arrows(4));
dot(pic2,(-5,3)); dot(pic2,(-4,5)); dot(pic2,(-3,1)); dot(pic2,(-2,0));
dot(pic2,(-1,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(2,-2));
label(pic1,"(2)",(0,-9));
label(pic2,"(3)",(0,-9));
label(pic3,"(4)",(0,-9));
label(pic4,"(5)",(0,-9));
add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy] | 60 | 7/8 |
Given two tangent circles $k$ and $k_{1}$ with a radius of one unit each in a plane. One of their common external tangents is the line $e$. Subsequently, we draw circles $k_{2}, k_{3}, \ldots, k_{n}$ such that each of them is tangent to $k$, $e$, and the circle with a sequence number one less than its own. What is the radius of the circle $k_{n}$? | \frac{1}{n^{2}} | 4/8 |
The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 12? | 36 | 7/8 |
Person A and Person B start simultaneously from points A and B respectively, walking towards each other. Person A starts from point A, and their speed is 4 times that of Person B. The distance between points A and B is \( S \) kilometers, where \( S \) is a positive integer with 8 factors. The first time they meet at point C, the distance \( AC \) is an integer. The second time they meet at point D, the distance \( AD \) is still an integer. After the second meeting, Person B feels too slow, so they borrow a motorbike from a nearby village near point D. By the time Person B returns to point D with the motorbike, Person A has reached point E, with the distance \( AE \) being an integer. Finally, Person B chases Person A with the motorbike, which travels at 14 times the speed of Person A. Both arrive at point A simultaneously. What is the distance between points A and B?
\[ \text{The distance between points A and B is } \qquad \text{kilometers.} \] | 105 | 1/8 |
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is | 16 | 1/8 |
Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h. | \text{No} | 1/8 |
The state income tax where Kristin lives is levied at the rate of $p\%$ of the first
$\textdollar 28000$ of annual income plus $(p + 2)\%$ of any amount above $\textdollar 28000$. Kristin
noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her
annual income. What was her annual income?
$\textbf{(A)}\,\textdollar 28000 \qquad \textbf{(B)}\,\textdollar 32000 \qquad \textbf{(C)}\,\textdollar 35000 \qquad \textbf{(D)}\,\textdollar 42000 \qquad \textbf{(E)}\,\textdollar 56000$ | \textbf{(B)}\,\textdollar32000 | 1/8 |
Find the radius of the circle inscribed in triangle $PQR$ if $PQ = 26$, $PR=10$, and $QR=18$. Express your answer in simplest radical form. | \sqrt{17} | 1/8 |
Natural numbers \(a\) and \(b\) are such that \(2a + 3b = \operatorname{lcm}(a, b)\). What values can the number \(\frac{\operatorname{lcm}(a, b)}{a}\) take? List all possible options in ascending or descending order separated by commas. If there are no solutions, write the number 0. | 0 | 3/8 |
For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$. | 512 | 1/8 |
The edge length of cube $ABCD A_1 B_1 C_1 D_1$ is 1. On the extension of edge $AD$ beyond point $D$, point $M$ is chosen such that $|AM| = 2\sqrt{2/5}$. Point $E$ is the midpoint of edge $A_1 B_1$, and point $F$ is the midpoint of edge $D D_1$. What is the maximum value of the ratio $|MP|/|PQ|$, where point $P$ lies on segment $AE$, and point $Q$ lies on segment $CF$? | \sqrt{2} | 5/8 |
Write in ascending order the multiples of 3 which, when 1 is added, are perfect squares, i.e., $3, 15, 24, 48, \ldots$ What is the multiple of 3 in the $2006^{\mathrm{th}}$ position? | 9060099 | 7/8 |
Given a list of the first 12 positive integers such that for each $2\le i\le 12$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list, calculate the number of such lists. | 2048 | 6/8 |
Given \( x \in \mathbb{R} \), find the maximum value of \(\frac{\sin x(2-\cos x)}{5-4 \cos x}\). | \frac{\sqrt{3}}{4} | 5/8 |
In an $n$ -by- $m$ grid, $1$ row and $1$ column are colored blue, the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n,m)$ | 96 | 7/8 |
Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits. | 397 | 7/8 |
Given that four integers \( a, b, c, d \) are all even numbers, and \( 0 < a < b < c < d \), with \( d - a = 90 \). If \( a, b, c \) form an arithmetic sequence and \( b, c, d \) form a geometric sequence, then find the value of \( a + b + c + d \). | 194 | 6/8 |
The square of a natural number \( a \) gives a remainder of 8 when divided by a natural number \( n \). The cube of the number \( a \) gives a remainder of 25 when divided by \( n \). Find \( n \). | 113 | 7/8 |
A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01? | 11475 | 6/8 |
Find the largest possible number of rooks that can be placed on a $3n \times 3n$ chessboard so that each rook is attacked by at most one rook. | 4n | 1/8 |
From the 4 internal angles of a quadrilateral, take any 2 angles and sum them. There are 6 possible sums. What is the maximum number of these sums that can be greater than $180^{\circ}$? | 3 | 2/8 |
Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients. | f(x)=x | 1/8 |
Express \((2207 - \frac{1}{2207 - \frac{1}{2207 - \frac{1}{2207 - \cdots}}})^{\frac{1}{8}}\) in the form \(\frac{a + b\sqrt{c}}{d}\), where \(a\), \(b\), \(c\), and \(d\) are integers. | \frac{3+\sqrt{5}}{2} | 1/8 |
Six soccer teams are competing in a tournament in Waterloo. Every team is to play three games, each against a different team. How many different schedules are possible? | 70 | 3/8 |
We are given a positive integer $s \ge 2$ . For each positive integer $k$ , we define its *twist* $k’$ as follows: write $k$ as $as+b$ , where $a, b$ are non-negative integers and $b < s$ , then $k’ = bs+a$ . For the positive integer $n$ , consider the infinite sequence $d_1, d_2, \dots$ where $d_1=n$ and $d_{i+1}$ is the twist of $d_i$ for each positive integer $i$ .
Prove that this sequence contains $1$ if and only if the remainder when $n$ is divided by $s^2-1$ is either $1$ or $s$ . | n\equiv1 | 2/8 |
Tourists Vitya and Pasha are traveling from city A to city B at the same speed, and tourists Katya and Masha are traveling from city B to city A at the same speed. Vitya meets Masha at 12:00, Pasha meets Masha at 15:00, and Vitya meets Katya at 14:00. How many hours after noon do Pasha and Katya meet? | 5 | 6/8 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x + y) + xf(y) - xy - x + 1\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(1),$ and let $s$ be the sum of all possible values of $f(1).$ Find $n \times s.$ | 2 | 7/8 |
The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there? | 432 | 3/8 |
The line $L_{1}$: $ax+(1-a)y=3$ and $L_{2}$: $(a-1)x+(2a+3)y=2$ are perpendicular to each other, find the values of $a$. | -3 | 5/8 |
Show that for some \( t > 0 \), we have \( \frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} + \frac{1}{1+d} > t \) for all positive \( a, b, c, d \) such that \( abcd = 1 \). Find the smallest such \( t \). | 1 | 6/8 |
Given a parabola \(C\) with the center of ellipse \(E\) as its focus, the parabola \(C\) passes through the two foci of the ellipse \(E\), and intersects the ellipse \(E\) at exactly three points. Find the eccentricity of the ellipse \(E\). | \frac{2 \sqrt{5}}{5} | 3/8 |
Subset \( S \subseteq \{1, 2, 3, \ldots, 1000\} \) is such that if \( m \) and \( n \) are distinct elements of \( S \), then \( m + n \) does not belong to \( S \). What is the largest possible number of elements in \( S \)? | 501 | 4/8 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 2} \frac{1-2^{4-x^{2}}}{2\left(\sqrt{2 x}-\sqrt{3 x^{2}-5 x+2}\right)}
$$ | -\frac{8\ln2}{5} | 6/8 |
In the two-dimensional Cartesian coordinate system \(xOy\), the circle \(C_{1}: x^{2}+y^{2}-a=0\) is symmetric with respect to the line \(l\) to the circle \(C_{2}: x^{2}+y^{2}+2x-2ay+3=0\). Determine the equation of the line \(l\). | 2x-4y+5=0 | 7/8 |
Given the function
$$
f(x)=\sin \left(\omega x+\frac{\pi}{4}\right) \quad (\omega>0)
$$
which has a maximum value but no minimum value on the interval $\left(\frac{\pi}{12}, \frac{\pi}{3}\right)$, determine the range of values for $\omega$. | (\frac{3}{4},3) | 2/8 |
A game of drawing balls involves a non-transparent paper box containing $6$ identical-sized, differently colored glass balls. Participants pay $1$ unit of fee to play the game once, drawing balls with replacement three times. Participants must specify a color from the box before drawing. If the specified color does not appear, the game fee is forfeited. If the specified color appears once, twice, or three times, the participant receives a reward of $0$, $1$, or $k$ times the game fee ($k \in \mathbb{N}^{*}$), respectively, and the game fee is refunded. Let $X$ denote the participant's profit per game in units of the fee.
(1) Calculate the value of the probability $P(X=0)$;
(2) Determine the minimum value of $k$ such that the expected value of the profit $X$ is not less than $0$ units of the fee.
(Note: Probability theory originated from gambling. Please consciously avoid participating in improper games!) | 110 | 7/8 |
Given an acute-angled triangle \(A_{0} B_{0} C_{0}\). Let the points \(A_{1}, B_{1}, C_{1}\) be the centers of the squares constructed on the sides \(B_{0} C_{0}, C_{0} A_{0}, A_{0} B_{0}\). We repeat the same construction with triangle \(A_{1} B_{1} C_{1}\) to obtain triangle \(A_{2} B_{2} C_{2}\), and so on. Prove that \(\Delta A_{n+1} B_{n+1} C_{n+1}\) intersects \(\Delta A_{n} B_{n} C_{n}\) exactly at 6 points. | 6 | 3/8 |
The isosceles triangle and the square shown here have the same area in square units. What is the height of the triangle, $h$, in terms of the side length of the square, $s$?
[asy]
draw((0,0)--(0,10)--(10,10)--(10,0)--cycle);
fill((0,0)--(17,5)--(0,10)--cycle,white);
draw((0,0)--(17,5)--(0,10)--cycle);
label("$s$",(5,10),N);
label("$h$",(6,5),N);
draw((0,5)--(17,5),dashed);
draw((0,5.5)--(0.5,5.5)--(0.5,5));
[/asy] | 2s | 6/8 |
The distances between the points are given as $A B = 30$, $B C = 80$, $C D = 236$, $D E = 86$, $E A = 40$. What is the distance $E C$? | 150 | 1/8 |
On a blackboard, the number 123456789 is written. Select two adjacent digits from this number, and if neither of them is 0, subtract 1 from each and swap their positions. For example: \( 123456789 \rightarrow 123436789 \rightarrow \cdots \). After performing this operation several times, what is the smallest possible number that can be obtained? The answer is __. | 101010101 | 1/8 |
Two ascetics live on the top of a vertical cliff of height \( h \) and at a distance from a neighboring village \( m \) times greater. One ascetic descends the cliff and then walks directly to the village. The other ascetic ascends to a certain height \( x \) and then flies directly to the village. If both of them travel the same distance, what is the height \( x \) that the second ascetic ascends to? | \frac{}{+2} | 4/8 |
Dima's mother told him he needed to eat 13 spoons of porridge. Dima told his friend that he ate 26 spoons of porridge. Each subsequent child, when talking about Dima's feat, increased the number of spoons by 2 or 3 times. Eventually, one of the children told Dima's mother that Dima ate 33,696 spoons of porridge. How many times in total, including Dima, did the children talk about Dima's feat? | 9 | 4/8 |
For the set $M$, define the function $f_M(x) = \begin{cases} -1, & x \in M \\ 1, & x \notin M \end{cases}$. For two sets $M$ and $N$, define the set $M \triangle N = \{x | f_M(x) \cdot f_N(x) = -1\}$. Given $A = \{2, 4, 6, 8, 10\}$ and $B = \{1, 2, 4, 8, 16\}$.
(1) List the elements of the set $A \triangle B = \_\_\_\_\_$;
(2) Let $\text{Card}(M)$ represent the number of elements in a finite set $M$. When $\text{Card}(X \triangle A) + \text{Card}(X \triangle B)$ takes the minimum value, the number of possible sets $X$ is $\_\_\_\_\_$. | 16 | 5/8 |
Let $AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$ . Let $B$ and $D$ be points chosen on segments $AE$ and $AF,$ respectively, such that $BD$ is parallel to $EF.$ Point $C$ is chosen in the interior of triangle $AEF$ such that $ABCD$ is cyclic. If $BC = 3$ and $CD = 4,$ then the ratio of areas $\tfrac{[ABCD]}{[AEF]}$ can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a, b$ . Compute $100a + b$ . | 5300 | 1/8 |
What is the area of the region enclosed by $x^2 + y^2 = |x| - |y|$? | \frac{\pi}{2} | 1/8 |
A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of $2\sqrt{2}$ from its original location?
*Proposed by Connor Gordon* | 3/8 | 6/8 |
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over $80$. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over $80$, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of $30$ multiple choice problems and that one's score, $s$, is computed by the formula $s=30+4c-w$, where $c$ is the number of correct answers and $w$ is the number of wrong answers. (Students are not penalized for problems left unanswered.)
| 119 | 1/8 |
Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n) $$ has no solution in integers positive $(m,n)$ with $m\neq n$ .
| 2 | 1/8 |
In four classes of a school, there are more than 70 children, and all of them attended the parallel meeting (no other children were present at the meeting).
Each girl who came was asked: "How many people from your class came, including you?"
Each boy who came was asked: "How many boys from your class came, including you?"
Among the responses were the numbers $7, 9, 10, 12, 15, 16, 19,$ and $21$ (all children answered correctly).
(a) How many children study in the largest class of the parallel?
(b) How many girls attended the parallel meeting? | 33\, | 1/8 |
In triangle \(ABC\), \(AB = 28\), \(BC = 21\), and \(CA = 14\). Points \(D\) and \(E\) are on \(AB\) with \(AD = 7\) and \(\angle ACD = \angle BCE\). Find the length of \(BE\). | 12 | 6/8 |
Let \( n \) be a positive integer not exceeding 1996. If there exists a \( \theta \) such that \( (\sin \theta + i \cos \theta)^{n} = \sin \theta + i \cos n \theta \), find the number of possible values for \( n \). | 499 | 1/8 |
Let $x_1=97$, and for $n>1$, let $x_n=\frac{n}{x_{n-1}}$. Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$. | 384 | 6/8 |
Currently, 7 students are to be assigned to participate in 5 sports events, with the conditions that students A and B cannot participate in the same event, each event must have at least one participant, and each student can only participate in one event. How many different ways can these conditions be satisfied? (Answer with a number) | 15000 | 7/8 |
In an addition problem where the digits were written on cards, two cards were swapped, resulting in an incorrect expression: $37541 + 43839 = 80280$. Find the error and write the correct value of the sum. | 81380 | 1/8 |
Consider the following 27 points of a cube: the center (1), the centers of the faces (6), the vertices (8), and the centers of the edges (12). We color each of these points either blue or red. Can this be done in such a way that there are no three points of the same color aligned along a straight line? Prove your answer. | No | 1/8 |
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