prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$? | 28 |
The product $55 \cdot 60 \cdot 65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number among these 5 numbers. | 20 |
$N$ is a 50-digit number in decimal representation. All digits except the 26th digit (from the left) are 1. If $N$ is divisible by 13, find its 26th digit. | 3 |
A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer? | 13 |
For each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n! + 1$ and $(n+1)!$. For $n < 100$, find the largest value of $h_n$. | 97 |
Each of the 2009 distinct points in the plane is colored either blue or red. On every unit circle centered at a blue point, there are exactly two red points. Find the greatest possible number of blue points. | 1964 |
Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , \ldots, \mathbb{S}_n$ with the following properties: 1. $|\mathbb{S}_i \cup \mathbb{S}_j| \leq 2004$ for any two indices $1 \leq i, j \leq n$. 2. $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1, 2, \ldots, ... | 32 |
Princess Pear has $100$ jesters with heights $1, 2, \dots, 100$ inches. On day $n$ with $1 \leq n \leq 100$, Princess Pear holds a court with all her jesters with height at most $n$ inches, and she receives two candied cherries from every group of $6$ jesters with a median height of $n - 50$ inches. A jester can be par... | 384160000 |
Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$. | 2077 |
From a list of integers from $1$ to $2022$, inclusive, delete all numbers in which at least one of its digits is a prime. How many numbers remain without erasing? | 431 |
We say that a positive integer $m$ covers the number $1998$ if the digits $1, 9, 9, 8$ appear in this order as digits of $m$. For instance, $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$. Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n ... | 1 |
The sum \[ \frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!} \] can be expressed as a rational number \(N\). Find the last 3 digits of \(2021! \cdot N\). | 977 |
How many positive integers less than 2005 are relatively prime to 1001? | 1442 |
There are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case. | 2017 |
Let $f(n) = 1 \times 3 \times 5 \times \cdots \times (2n-1)$. Compute the remainder when $f(1) + f(2) + f(3) + \cdots + f(2016)$ is divided by $100$. | 24 |
Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) \cdots (1 - 2008x)$. | -1004 |
The integer $n$, between 10000 and 99999, is represented as $abcde$ in decimal notation. The digits $a$, $b$, $c$, $d$, and $e$ are defined as follows: - $a$ is the remainder when $n$ is divided by 2. - $b$ is the remainder when $n$ is divided by 3. - $c$ is the remainder when $n$ is divided by 4. - $d$ is the remainde... | 11311 |
Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$. | 167 |
In the $3$-dimensional coordinate space, find the distance from the point $(36, 36, 36)$ to the plane that passes through the points $(336, 36, 36)$, $(36, 636, 36)$, and $(36, 36, 336)$. | 200 |
How many solutions does the equation \( \frac{\lfloor x \rfloor}{\{ x \}} = \frac{2007x}{2008} \) have? | 2006 |
$\text{HOW}$, $\text{BOW}$, and $\text{DAH}$ are equilateral triangles in a plane such that $WO = 7$ and $AH = 2$. Given that $D$, $A$, $B$ are collinear in that order, find the length of $BA$. | 11 |
In how many ways can each of the integers $1$ through $11$ be assigned one of the letters $L$, $M$, and $T$ such that consecutive multiples of $n$, for any positive integer $n$, are not assigned the same letter? | 24 |
Find the least positive integer $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$. | 556 |
A $1 \times 5$ rectangle is divided into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at cell 1. Every second, it jumps from its current cell to one of the adjacent cells. The frog makes exactly 14 jumps. How many paths can the frog take to finish at cell 5? | 364 |
Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0 \leq n < 16} 2^n (-1)^{s(n)}. \] | 45 |
In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ columns and $4$ rows such that the $16$ squares of the intersections are all of the same color. Find th... | 601 |
Let $ABC$ be a triangle and $D$, $E$, and $F$ be the midpoints of sides $BC$, $AC$, and $BC$ respectively. The medians $AD$ and $BE$ are perpendicular, with $AD = 12$ and $BE = 9$. What is the value of $CF$? | 15 |
Positive numbers are written in the squares of a $10 \times 10$ table. Frogs sit in five squares and cover the numbers in these squares. Kostya found the sum of all visible numbers and got 10. Then each frog jumped to an adjacent square and Kostya’s sum changed to $10^2$. Then the frogs jumped again, and the sum change... | 1000000 |
For each positive integer $k$, denote $C(k)$ as the sum of its distinct prime divisors. For example, $C(1) = 0$, $C(2) = 2$, and $C(45) = 8$. Find all positive integers $n$ for which $C(2^n + 1) = C(n)$. | 3 |
Find all real numbers $x$ such that $4x^5 - 7$ and $4x^{13} - 7$ are both perfect squares. | 2 |
Denote by $s(x)$ the sum of the digits of a positive integer x. Solve: x + s(x) + s(s(x)) + s(s(s(x))) = 1993$ | 1963 |
Javiera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next, Javiera says a natural number $k$, where $1 \leq k \leq n$, and Claudio must move the token to the right or to the left by $k$ squares, at his choice, and so on. Javiera wins if she ... | 1010 |
Given that $x$, $y$ are positive integers such that $x(x+1)$ divides $y(y+1)$, but neither $x$ nor $x+1$ divides either $y$ or $y+1$. Find the smallest possible value of $x^2 + y^2$. | 596 |
Determine all possible values of the expression $xy + yz + zx$ given that $x, y, z$ are real numbers satisfying the conditions $x^2 - yz = y^2 - zx = z^2 - xy = 2$. | -2 |
Find all prime numbers $p$ such that $5^p + 4p^4$ is a perfect square. | 5 |
In a room, there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you turn on the $n$-th switch, the $n$-th bulb will light up. There is a group of men who are operating the switches according to the following rule: - They enter one by one and start flipping the switches starting from ... | 1024 |
The kid cut out a rectangle from grid paper, with each cell having a side length of 1, and calculated its area and perimeter. Carlson then took the scissors and cut out a square from this rectangle, along the grid lines, that was adjacent to the boundary of the rectangle. - "My rectangle..." the kid sobbed. "There is s... | 2 |
Jay has a $24 \times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e., it switches lights that are on to off and lights that are off to on. Jay toggles each of the $48$ rows and columns exactly once,... | 9408 |
Consider a graph $G$ with 2019 nodes. Define "inverting" a vertex $v$ as follows: for every other vertex $u$, if there was an edge between $v$ and $u$, it is deleted; if there wasn't, it is added. We aim to minimize the number of edges in the graph through several inverting operations (we can invert the same vertex mul... | 1018081 |
The number $125$ can be expressed as a sum of pairwise coprime integers, each greater than $1$. Determine the maximum number of terms possible in such a sum. | 8 |
Does there exist a polynomial of one variable that is irreducible over the field of integers, has degree $60$, and is a multiple of the form $X^n - 1$? If so, how many such polynomials exist? | 9 |
Find all positive integers $N$ that have at least 4 positive divisors such that the sum of the squares of the 4 smallest positive divisors of $N$ equals $N$. | 130 |
Determine the smallest natural number $n$ such that among any $n$ integers, one can choose $18$ integers whose sum is divisible by $18$. | 35 |
A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times. Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum numb... | 18 |
Two linear functions $f(x)$ and $g(x)$ satisfy the following properties for all $x$: - $f(x) + g(x) = 2$ - $f(f(x)) = g(g(x))$ Additionally, $f(0) = 2022$. Compute $f(1)$. | 1 |
The value of the expression $$\sum_{n=2}^{\infty} \frac{\binom{n}{2}}{7^{n-2}} = 1+\frac{3}{7}+\frac{6}{49}+\frac{10}{343}+\frac{15}{2401}+\dots+\frac{\binom{n}{2}}{7^{n-2}}+\cdots$$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 559 |
Find the sum of all integers $n$ satisfying the following inequality: \[ \frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}. \] | 33 |
A natural number $n$ is called perfect if it is equal to the sum of all its natural divisors other than $n$. For example, the number $6$ is perfect because $6 = 1 + 2 + 3$. Find all even perfect numbers that can be expressed as the sum of two cubes of positive integers. | 28 |
There are $10001$ students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: 1. Each ... | 5000 |
If positive reals $x, y$ satisfy the equation $2(x+y) = 1 + xy$, find the minimum value of the expression $$A = x + \frac{1}{x} + y + \frac{1}{y}.$$ | 8 |
Let $a$ and $b$ be real numbers such that $$\left(8^a+2^{b+7}\right)\left(2^{a+3}+8^{b-2}\right)=4^{a+b+2}.$$ The value of the product $ab$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 271 |
Let $S$ be the solid in three-dimensional space consisting of all points ( $x, y, z$ ) satisfying the following system of six simultaneous conditions: $$ \begin{aligned} & x \geqslant 0, \quad y \geqslant 0, \quad z \geqslant 0 \\ & x+y+z<11, \\ & 2 x+4 y+3 z \leqslant 36 \\ & 2 x+3 z \leqslant 24. \end{aligned} $$ Det... | 7 |
Suppose $a$ and $b$ are real numbers such that the polynomial $x^3 + ax^2 + bx + 15$ has a factor of $x^2 - 2$. Find the value of $a^2b^2$. | 225 |
Let $S$ be the solid in three-dimensional space consisting of all points ( $x, y, z$ ) satisfying the following system of six simultaneous conditions: $$ \begin{aligned} & x \geqslant 0, \quad y \geqslant 0, \quad z \geqslant 0 \\ & x+y+z<11, \\ & 2 x+4 y+3 z \leqslant 36 \\ & 2 x+3 z \leqslant 24. \end{aligned} $$ Det... | 11 |
Find all integers $n$ such that $n^{n-1} - 1$ is square-free. | 2 |
Determine all positive integers $n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y,z) = (x - y)^{5n} + (y - z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients. | 1 |
Determine all natural numbers $n$ such that $9^n - 7$ can be represented as a product of at least two consecutive natural numbers. | 1 |
On a circumference of a circle, seven points are selected, at which different positive integers are assigned to each of them. Then simultaneously, each number is replaced by the least common multiple of the two neighboring numbers to it. If the same number $n$ is obtained in each of the seven points, determine the smal... | 420 |
Let $f(x) = 5x^{13} + 13x^5 + 9ax$. Find the least positive integer $a$ such that $65$ divides $f(x)$ for every integer $x$. | 63 |
For what integers $n \geq 3$ is it possible to arrange the numbers $1, 2, \ldots, n$ in a circular order such that every number divides the sum of the next two numbers, in a clockwise direction? | 3 |
In a circle with radius $10$, point $M$ is located on the chord $PQ$ such that $PM = 5$ and $MQ = 10$. Through point $M$, we draw chords $AB$ and $CD$. Points $X$ and $Y$ are the intersection points of chords $AD$ and $BC$ with chord $PQ$, respectively. Given that $XM = 3$, find the length of $MY$. | 2 |
Determine all integers $n > 1$ whose positive divisors add up to a power of $3$. | 2 |
The nonzero real numbers $a, b, c$ satisfy the following system of equations: \[ \begin{cases} a + ab = c \\ b + bc = a \\ c + ca = b \end{cases} \] Find all possible values of the product $abc$. | 3 |
A positive integer $n$ is considered "funny" if for every positive divisor $d$ of $n$, the number $d+2$ is a prime number. Determine all funny numbers that have the maximum possible number of divisors. | 135 |
Let $n$ be a positive integer such that no matter how $10^n$ is expressed as the product of two positive integers, at least one of these two integers contains the digit 0. Find the smallest possible value of $n$. | 8 |
In rhombus $ABCD$, let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. If $CN = 7$ and $DM = 24$, compute $AB^2$. | 250 |
Joana divided \(365\) by all integers from \(1\) to \(365\) and added all the remainders. Then she divided \(366\) by all the integers from \(1\) to \(366\) and also added all the remainders. Which of the two sums is greater and what is the difference between them? | 13 |
Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$. Find $b - a$. | 223 |
For a given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define the function recursively as $f_{n+1}(k) = f_1(f_n(k))$. Determine the value of $f_{1991}(2^{1990})$. | 256 |
Determine the minimum possible amount of distinct prime divisors of $19^{4n} + 4$, where $n$ is a positive integer. | 3 |
Let $p, q$ be prime numbers such that $n^{3pq} - n$ is a multiple of $3pq$ for all positive integers $n$. Find the least possible value of $p + q$. | 28 |
Let $m, n, p$ be fixed positive real numbers which satisfy $m n p=8$. Depending on these constants, find the minimum of $$ x^{2}+y^{2}+z^{2}+m x y+n x z+p y z, $$ where $x, y, z$ are arbitrary positive real numbers satisfying $x y z=8$. When is the equality attained? Solve the problem for: $m=n=p=2$ | 36 |
A set of positive integers is said to be *pilak* if it can be partitioned into two disjoint subsets $F$ and $T$, each with at least two elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the set ... | 30 |
More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors lost exactly two games each. All other competitors won exactly three games each. There were no draws in the tournament. Determine how many competitors there w... | 12 |
For any natural number, let $S(n)$ denote the sum of the digits of $n$. Find the number of 3-digit numbers for which $S(S(n)) = 2$. | 100 |
What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$? | 20 |
Let $a_n=6^{n}+8^{n}$ . Determine the remainder upon dividing $a_ {83}$ by $49$ . | 35 |
Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in l... | 57 |
What is the largest $2$ -digit prime factor of the integer $n = {200\choose 100}$ ? | 61 |
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 |
Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. [asy] draw(circle((0,0),4)); draw((-4,0)--(4,0)); draw... | 65 |
For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+... | 448 |
In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$. [asy]size(160); defaultpen(linewidth(.8pt)+fontsize(... | 130 |
The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is $5$, that $BC=6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the central angle o... | 175 |
Find the value of $a_2+a_4+a_6+a_8+\ldots+a_{98}$ if $a_1$, $a_2$, $a_3\ldots$ is an arithmetic progression with common difference 1, and $a_1+a_2+a_3+\ldots+a_{98}=137$. | 93 |
Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$? | 649 |
Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the ab... | 24 |
The function $f$ is defined on the set of integers and satisfies $f(n)=\begin{cases} n-3&{if}\ n\ge 1000\\ f(f(n+5))&{if}\ n<1000\end{cases}$ Find $f(84)$ . | 997 |
The equation $z^6+z^3+1=0$ has complex roots with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane . Determine the degree measure of $\theta$ . | 160 |
In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is $15\mbox{cm}^2$ and the area of face $ABD$ is $12 \mbox { cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\mbox{cm}^3$. | 20 |
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... | 119 |
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$ . | 106 |
A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$ . If $x=0$ is a root for $f(x)=0$ , what is the least number of roots $f(x)=0$ must have in the interval $-1000\leq x \leq 1000$ ? | 401 |
Determine $x^2+y^2+z^2+w^2$ if $\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$ $\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$ $\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$ $\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\fra... | 36 |
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 |
A sequence of integers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492? | 986 |
The sum of the following seven numbers is exactly $19$: $a_1 = 2.56$ , $a_2 = 2.61$ , $a_3 = 2.65$ , $a_4 = 2.71$ , $a_5 = 2.79$ , $a_6 = 2.82$ , $a_7 = 2.86$ . It is desired to replace each $a_i$ by an integer approximation $A_i$ , $1\le i \le 7$ , so that the sum of the $A_i$ 's is also $19$ and so that $M$ , the max... | 61 |
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator... | 49 |
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis? | 85 |
Let $A$, $B$, $C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures $1$ meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to th... | 182 |
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