prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Let $a, b, c, x, y,$ and $z$ be complex numbers such that \[ a = \frac{b+c}{x-2}, \ b = \frac{c+a}{y-2}, \ c = \frac{a+b}{z-2}. \] If $xy + yz + xz = 67$ and $x + y + z = 2010$, find the value of $xyz$. | -5892 |
Given 2005 distinct numbers $a_1, a_2, \dots, a_{2005}$. By one question, we may take three different indices $1 \le i < j < k \le 2005$ and find out the set of numbers $\{a_i, a_j, a_k\}$ (unordered, of course). Find the minimal number of questions necessary to determine all numbers $a_i$. | 1003 |
Point $P_1$ is located 600 miles West of point $P_2$. At 7:00 AM, a car departs from $P_1$ and drives East at a speed of 50 mph. At 8:00 AM, another car departs from $P_2$ and drives West at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of... | 60 |
Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $$\{10a+b \mid 1 \leq a \leq 5, 1 \leq b \leq 5\}$$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$... | 47 |
Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le 3$, let $D_i$, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$. Suppose that the areas of the triangles $A_1A_2A_... | 917 |
Let $P$ be the product of all non-zero digits of the positive integer $n$. For example, $P(4) = 4$, $P(50) = 5$, $P(123) = 6$, $P(2009) = 18$. Find the value of the sum: $P(1) + P(2) + \ldots + P(2008) + P(2009)$. | 194763 |
There is a sequence defined as follows: - $a(2) = 0$ - $a(3) = 1$ - For $n \geq 4$, $a(n) = a\left(\left\lfloor \frac{n}{2} \right\rfloor\right) + a\left(\left\lceil \frac{n}{2} \right\rceil\right)$ Find $a(2014)$. Note: $\left\lfloor \frac{n}{2} \right\rfloor$ and $\left\lceil \frac{n}{2} \right\rceil$ denote the floo... | 34 |
Let $a_k$ be the number of ordered $10$-tuples $(x_1, x_2, \ldots, x_{10})$ of nonnegative integers such that: \[ x_1^2 + x_2^2 + \ldots + x_{10}^2 = k. \] Define $b_k = 0$ if $a_k$ is even and $b_k = 1$ if $a_k$ is odd. Determine the value of: \[ \sum_{i=1}^{2012} b_{4i}. \] | 1006 |
A right triangle has the property that its sides are pairwise relatively prime positive integers and that the ratio of its area to its perimeter is a perfect square. Find the minimum possible area of this triangle. | 30 |
Let \( x \) be the answer to this question. Find the value of \( 2017 - 2016x \). | 1 |
Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament. They agreed on the following rules of attendance: 1. There should always be at least one person present on each day. 2. On no two days should the same subset of politicians attend. 3. The members present on day $N$ sh... | 128 |
The value of the infinite series $$\sum_{n=2}^\infty \dfrac{n^4+n^3+n^2-n+1}{n^6-1}$$ can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive numbers. Compute $p+q$. | 23 |
Let $O$ be the circumcenter of triangle $ABC$ with circumradius $15$. Let $G$ be the centroid of $\triangle ABC$ and let $M$ be the midpoint of $BC$. If $BC = 18$ and $\angle MOA = 150^\circ$, find the area of $\triangle OMG$. | 15 |
In a simple graph with 300 vertices, no two vertices of the same degree are adjacent. What is the maximal possible number of edges in such a graph? | 42550 |
Find the number of positive integers $n$ such that $1 \leq n \leq 1000$ and $n$ is divisible by $\lfloor \sqrt[3]{n} \rfloor$. | 172 |
A graph has $30$ points, and each point has $6$ edges. Find the total number of triples such that each pair of points is either joined or not joined. | 4060 |
Find the integer part of the number \( \frac{2^1}{1!} + \frac{2^2}{2!} + \cdots + \frac{2^{2018}}{2018!} \). | 6 |
Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that for all $x, y \in \mathbb{R}^+$, $f(x)f(y) = f(xy) + f\left( \frac{x}{y}\right)$, where $\mathbb{R}^+$ represents the positive real numbers. Given that $f(2) = 3$, compute the last two digits of $f(2^{2^{2020}})$. | 47 |
What is the greatest three-digit divisor of $111777$? | 703 |
Two types of pieces, bishops and rooks, are to be placed on a $10 \times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to attack a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that d... | 50 |
For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$? | 205 |
Evaluate $\int_{1}^{4} \frac{x-2}{(x^2+4)\sqrt{x}}\,dx$ | 0 |
On each non-boundary unit segment of an $8 \times 8$ chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers? | 0 |
Determine the smallest positive integer $n$ with the following property: for every triple of positive integers $x, y, z$, with $x$ dividing $y^3$, $y$ dividing $z^3$, and $z$ dividing $x^3$, it also holds that $(xyz)$ divides $(x + y + z)^n$. | 13 |
What is the greatest positive integer $m$ such that $n^2(1+n^2-n^4) \equiv 1 \pmod{2^m}$ for all odd integers $n$? | 7 |
Consider a box with dimensions $10\, \text{cm} \times 16\, \text{cm} \times 1\, \text{cm}$. Determine the maximum number of balls with a diameter of $1\, \text{cm}$ that the box can contain. | 160 |
A square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates $2$, $0$, and $18$ in some order. Find the sum of all possible values of $A$. | 1168 |
There are six lines in the plane. No two of them are parallel, and no point lies on more than three lines. What is the minimum possible number of points that lie on at least two lines? | 7 |
In trapezoid $ABCD$, $AD$ is parallel to $BC$, with the side lengths $AB = 6$, $BC = 7$, $CD = 8$, and $AD = 17$. If the sides $AB$ and $CD$ are extended to meet at point $E$, find the measure of the angle at $E$ (in degrees). | 90 |
Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\substack{abc=2310 \\ a,b,c\in \mathbb{N}}} (a+b+c),$$ where $\mathbb{N}$ denotes th... | 49140 |
Find the number of odd three-digit positive integers $x$ satisfying $x^2 \equiv 1 \pmod{8}$ and the product of the digits of $x$ is odd. | 125 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$? | 167 |
On a table near the sea, there are $N$ glass boxes where $N < 2021$, each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, ... | 101 |
How many positive integers at most $420$ leave different remainders when divided by each of $5$, $6$, and $7$? | 250 |
Ria writes down the numbers $1, 2, \ldots, 101$ using red and blue pens. The largest blue number is equal to the number of numbers written in blue, and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with the red pen? | 68 |
Let $f$ be a real-valued function such that $4f(x) + x f\left(\frac{1}{x}\right) = x + 1$ for every positive real number $x$. Find $f(2014)$. | 403 |
There are 100 houses on the street, divided into 50 pairs. Each pair consists of houses directly opposite each other across the street. On the right side, houses have even numbers, while on the left side, houses have odd numbers. The numbers on both sides increase from the start to the end of the street but are not nec... | 199 |
You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the re... | 1 |
Let $X$ be the set of all even positive integers $n$ such that the measure of the angle of some regular polygon is $n$ degrees. Find the number of elements in $X$. | 16 |
Evaluate the following expression: \[ 0 - 1 - 2 + 3 - 4 + 5 + 6 + 7 - 8 + \ldots + 2000 \] The terms with minus signs are exactly the powers of two. | 1996906 |
A regular $n$-gon $P_1P_2\ldots P_n$ satisfies $\angle P_1P_7P_8 = 178^\circ$. Compute $n$. | 630 |
The vertices of Durer's favorite regular decagon in clockwise order are labeled as $D_1, D_2, D_3, \ldots, D_{10}$. Determine the angle between the diagonals $D_1D_3$ and $D_2D_5$. | 54 |
Archit and Ayush are walking around on the set of points $(x,y)$ for all integers $-1 \leq x,y \leq 1$. Archit starts at $(1,1)$ and Ayush starts at $(1,0)$. Each second, they move to another point in the set chosen uniformly at random among the points with distance $1$ away from them. If the probability that Archit go... | 7 |
An equilateral triangle is given. A point lies on the incircle of this triangle. If the smallest two distances from the point to the sides of the triangle are $1$ and $4$, the side length of this equilateral triangle can be expressed as $\frac{a\sqrt{b}}{c}$, where $(a, c) = 1$ and $b$ is not divisible by the square of... | 34 |
Find the largest positive integer $n$ such that the number $(2n)!$ ends with 10 more zeroes than the number $n!$. | 42 |
Two players play a game, starting with a pile of $N$ tokens. On each player's turn, they must remove $2^n$ tokens from the pile for some nonnegative integer $n$. If a player cannot make a move, they lose. For how many $N$ between $1$ and $2019$ (inclusive) does the first player have a winning strategy? | 1346 |
Suppose that $a_1 = 1$, and for all $n \ge 2$, \[ a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1. \] Furthermore, suppose that \[ b_n = a_1 + a_2 + \ldots + a_n \] for all $n$. If \[ b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k \] for some $k$, find $k$. | 2022 |
Let triangle $ABC$ have $\angle A = 70^\circ$, $\angle B = 60^\circ$, and $\angle C = 50^\circ$. Extend altitude $BH$ past $H$ to point $D$ so that $BD = BC$. Find $\angle BDA$ in degrees. | 50 |
Let $S(n)$ be the sum of digits for any positive integer $n$ (in decimal notation). Let $N=\sum_{k=10^{2003}}^{10^{2004}-1} S(k)$. Determine $S(N)$. | 27 |
An infinite sequence of integers, $a_0, a_1, a_2, \dots$, with $a_0 > 0$, has the property that for $n \ge 0$, $a_{n+1} = a_n - b_n$, where $b_n$ is the number having the same sign as $a_n$, but having the digits written in reverse order. For example, if $a_0 = 1210$, $a_1 = 1089$, and $a_2 = -8712$, etc. Find the smal... | 1012 |
You are given that $17! = 355687ab8096000$ for some digits $a$ and $b$. Find the two-digit number $\overline{ab}$ that is missing above. | 42 |
Let $P$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $P$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB = 20$ and $CD = 14$, compute the area of $ABCD$. | 140 |
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is monotonically bounded if $1 \leq a_i \leq i$ for all $1 \leq i \leq 2^{16}$. We say that a term $a_k$ in the sequence with $2 \leq k \leq 2^{16}-1$ is a mountain if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes out all ... | 49153 |
Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$. | -7 |
Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other. What is the maximum value of $m$? | 1012 |
For any integer $a$, let $f(a) = |a^4 - 36a^2 + 96a - 64|$. What is the sum of all values of $f(a)$ that are prime? | 22 |
For any positive integer $x$, let $f(x) = x^x$. Suppose that $n$ is a positive integer such that there exists a positive integer $m$ with $m \neq 1$ such that $f(f(f(m))) = m^{m^{n+2020}}$. Compute the smallest possible value of $n$. | 13611 |
Let $S$ be the sum of all positive integers $n$ such that \( \frac{3}{5} \) of the positive divisors of \( n \) are multiples of \( 6 \) and \( n \) has no prime divisors greater than \( 3 \). Compute \( \frac{S}{36} \). | 2345 |
Let $a$ and $b$ be randomly selected three-digit integers and suppose $a > b$. We say that $a$ is clearly bigger than $b$ if each digit of $a$ is larger than the corresponding digit of $b$. If the probability that $a$ is clearly bigger than $b$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, compute ... | 1061 |
Will the number $\frac{1}{1996}$ decrease or increase, and by how many times, if in the decimal notation of this number, the first non-zero digit after the decimal point is crossed out? | 50 |
Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 centered at each of these $2^n$ points. Let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior of a... | 10 |
Let $f(n) = 9n^5 - 5n^3 - 4n$. Find the greatest common divisor of $f(17), f(18), \ldots ,f(2009)$. | 120 |
In $\triangle ABC$, let point $D$ be the foot of the altitude from $A$ to $BC$. Suppose that $\angle A = 90^{\circ}$, $AB - AC = 5$, and $BD - CD = 7$. Find the area of $\triangle ABC$. | 150 |
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold: 1. $f(1) = 1.$ 2. $\frac{(x + y)}{2} < f(x + y) \le f(x) + f(y)$ for all $x, y \in \mathbb{N}$. 3. $f(4n + 1) < 2f(2n + 1)$ for all $n \ge 0.$ 4. $f(4n + 3) \le 2f(2n + 1)$ for all $n \ge 0.$ Find the sum of all possible values o... | 1012 |
Positive integers $a$, $b$, and $c$ satisfy the equation: \[ \frac{ab}{a - b} = c \] What is the largest possible value of $a + b + c$ that does not exceed 99? | 99 |
Find the number of 5-digit numbers $n$, such that every digit of $n$ is either $0$, $1$, $3$, or $4$, and $n$ is divisible by $15$. | 40 |
The function $g(x)$ is defined as $\sqrt{\frac{x}{2}}$ for all positive $x$. Calculate the value of the expression: $$g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right)$$ This expression can be expressed as $\cos(b)$ in degrees, where $0^\circ < b < 90^\circ$ and $b = \frac{p}{q}$... | 19 |
Sokal da is trying to find the largest positive integer $n$ such that when $n$ is transformed to base-7, it appears as twice its value in base-10. For example, $156$ is such a number because $156_{10} = 312_7$ and $312 = 2 \times 156$. Determine Sokal da's number. | 315 |
Point $P$ is taken inside the square $ABCD$ such that $BP + DP = 25$, $CP - AP = 15$, and $\angle ABP = \angle ADP$. What is the radius of the circumcircle of $ABCD$? | 10 |
Call an $8$-digit number \textit{cute} if its digits are a permutation of $1,2,\ldots,8$. For example, $23615478$ is \textit{cute} but $31234587$ is not. Find the number of $8$-digit \textit{cute} numbers that are divisible by $11$. | 4608 |
Dr. Lisi writes out the first $2008$ terms of an arithmetic progression that begins $-1776, -1765, -1754, \ldots$. Joshua computes the (positive) difference between the $1980^\text{th}$ term in the sequence and the $1977^\text{th}$ term in the sequence. What number does Joshua compute? | 33 |
Let $ABCD$ be a square with side length $8$. A second square $A_1B_1C_1D_1$ is formed by joining the midpoints of $AB$, $BC$, $CD$, and $DA$. A third square $A_2B_2C_2D_2$ is formed in the same way from $A_1B_1C_1D_1$, and a fourth square $A_3B_3C_3D_3$ from $A_2B_2C_2D_2$. Find the sum of the areas of these four squar... | 120 |
Let $\alpha$ and $\beta$ be positive integers such that: $$ \frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} . $$ Find the smallest possible value of $\beta$. | 23 |
Colour a $20000 \times 20000$ square grid using 2000 different colours, with one colour per square. Two squares are considered neighbours if they share a vertex. Define a path as a sequence of squares where each pair of successive squares are neighbours. Mark $k$ squares on the grid. For each unmarked square $x$, there... | 2000 |
Among the numbers $11n + 10^{10}$, where $1 \le n \le 10^{10}$ is an integer, how many are squares? | 44801 |
Within a group of $2009$ people, every two people have exactly one common friend. Find the least value of the difference between the person with the maximum number of friends and the person with the minimum number of friends. | 2006 |
Positive integers $x, y, z$ satisfy the equation $x^3 + xy + x^2 + xz + y + z = 301$. Compute $y + z - x$. | 1 |
Find the least positive integer $m$ such that $\text{lcm}(15, m) = \text{lcm}(42, m)$. Here $\text{lcm}(a, b)$ is the least common multiple of $a$ and $b$. | 70 |
Consider digits $A, B, C, D$, with $A \neq 0$, such that $ABCD = (CD)^2 - (AB)^2$. Compute the sum of all distinct possible values of $A + B + C + D$. | 21 |
Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that: \[ Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right... | 68 |
Find the sum of the prime factors of $67208001$, given that $23$ is one. | 781 |
Given that the polynomial $P(x) = x^5 - x^2 + 1$ has 5 roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$. | 5 |
Find the sum of all integers $n$ with $2 \le n \le 999$ that have the following property: if $x$ and $y$ are randomly selected without replacement from the set $\{ 1, 2, \dots, n \}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number. | 600 |
Find the smallest positive integer $k$ such that \[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] for some ordered triple of positive integers $(a,b,c)$. | 1297 |
Let $f$ be a polynomial such that, for all real numbers $x$, \( f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019 \). Compute \( f(2018) \). | -2019 |
The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 < m < 6.5 \cdot 10^9$. Find $m$. | 6490716149 |
In a game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What is the largest value of $n$ such that this statement is true regardless of the order in which ... | 2 |
Find the last digit of the number $$\frac{400!}{(200!)(2^{200})}$$ | 5 |
You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2 \times 1$ and $3 \times 1$. The playing field should be completely covered with these dominoes without any overlaps, and no domino should protrude beyond the playing field. Furthermore, all dominoes must be aligned in... | 257 |
Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM = MB = BP = 15$ and $AN = NC = CP = 25$. | 150 |
An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself. For example, $28$ is a perfect number since $28 = 1 + 2 + 4 + 7 + 14$. Let $n!$ denote the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$, where $n$ is a positive integer. An integer is a factorial if and only i... | 6 |
How many positive integers have a square less than $10^7$? | 3162 |
A square grid $100 \times 100$ is tiled in two ways: only with dominoes and only with $2 \times 2$ squares. What is the least number of dominoes that are entirely inside some $2 \times 2$ square? | 100 |
A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube. | 125 |
Let $P(X) = X^5 + 3X^4 - 4X^3 - X^2 - 3X + 4$. Determine the number of monic polynomials $Q(x)$ with integer coefficients such that $\frac{P(X)}{Q(X)}$ is a polynomial with integer coefficients. Note: A monic polynomial is one with leading coefficient $1$ (e.g., $x^3 - 4x + 5$ is monic, but $5x^3 - 4x^2 + 1$ or $x^2 + ... | 12 |
In a chess tournament, there are $100$ players. Each player plays one game with every other player. For a win, $1$ point is awarded; for a loss, $0$ points are given; and for a draw, both players receive $0.5$ points. Ion scored more points than any other player. Mihai lost only one game but accumulated fewer points th... | 50 |
Let $a_1$, $a_2$, \ldots, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\ell$, and $m$ from 1 through 2019, at least two of the numbers $a_i$, $a_j$, $a_k$, $a_\ell$, and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the given seq... | 8 |
Is there a whole number that becomes exactly $57$ times less than itself when its first digit is crossed out? | 7125 |
What is the largest prime factor of 8091? | 31 |
Let $k$ be an integer. If the equation $(x-1)\left|x+1\right|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there? | 4544 |
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