prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Let $u_0, u_1, u_2, \ldots$ be integers such that $u_0 = 100$; $u_{k+2} \geq 2 + u_k$ for all $k \geq 0$; and $u_{\ell+5} \leq 5 + u_\ell$ for all $\ell \geq 0$. Find all possible values for the integer $u_{2023}$. | 2123 |
There are $110$ guinea pigs for each of the $110$ species, arranged in a $110 \times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs are arranged, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species. | 11 |
Compute \( \gcd \left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right) \) | 4050 |
S is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides $abcdef$ for all members of $S$. | 24 |
Let $ABC$ be an isosceles triangle such that $AB = AC$ and $\angle A = 20^\circ$. Let $M$ be the foot of the altitude from $C$, and let $N$ be a point on the side $AC$ such that $CN = \frac{1}{2} BC$. Determine the measure of the angle $AMN$. | 60 |
1. Let $f(n) = n^2 + 6n + 11$ be a function defined on positive integers. Find the sum of the first three prime values $f(n)$ takes on. | 753 |
Determine the minimum value of a prime number $p > 3$ for which there is no natural number $n > 0$ such that $2^n + 3^n \equiv 0 \pmod{p}$. | 19 |
The eight points $A, B, \ldots, G, H$ lie on five circles as shown. Each of these letters is represented by one of the eight numbers $1, 2, \ldots, 7, 8$, replaced such that the following conditions are met: 1. Each of the eight numbers is used exactly once. 2. The sum of the numbers on each of the five circles is the ... | 8 |
We say that an ordered pair $(a,b)$ of positive integers with $a > b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there? | 16 |
Evaluate the product \(\left(1 + \frac{2}{3}\right)\left(1 + \frac{2}{4}\right)\left(1 + \frac{2}{5}\right) \cdots \left(1 + \frac{2}{98}\right).\) | 825 |
Let $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$. What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n$? | 15 |
Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $\frac{n}{2}$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine continues operating by automatically using its output as a new input, stopping immediately before it processes a numbe... | 67 |
An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence \[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \] for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$. | 3 |
Triangle $ABC$ has side lengths $AB=18$, $BC=36$, and $CA=24$. The circle $\Gamma$ passes through point $C$ and is tangent to segment $AB$ at point $A$. Let $X$, distinct from $C$, be the second intersection of $\Gamma$ with $BC$. Moreover, let $Y$ be the point on $\Gamma$ such that segment $AY$ is an angle bisector of... | 69 |
Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, \ldots, 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of the sum $P_A + P_B$. | 402 |
How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these? | 33 |
Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$. | 1007 |
12 knights are sitting at a round table. Each knight is an enemy with the two knights adjacent to them, but not with any of the others. We need to choose 5 knights to save the princess such that no two knights in the group are enemies. How many ways can this be done? | 36 |
In the figure below, parallelograms $ABCD$ and $BFEC$ have areas $1234\, \text{cm}^2$ and $2804\, \text{cm}^2$, respectively. Points $M$ and $N$ are chosen on sides $AD$ and $FE$, respectively, so that segment $MN$ passes through $B$. Find the area of $\triangle MNC$. | 2019 |
The regular octagon in the figure is inscribed in a circle with a radius of $1$. Let $P$ be an arbitrary point on this circle. Calculate the value of $PA^2 + PB^2 + \ldots + PH^2$. | 16 |
Let $ABCD$ be a square face of a cube with edge length $2$. A plane $P$ that contains point $A$ and the midpoint of line segment $\overline{BC}$ splits the cube into two pieces of equal volume. What is the square of the area of the intersection of plane $P$ and the cube? | 24 |
Let $\frac{a}{b}$ be a fraction such that $a$ and $b$ are positive integers and the first three digits of its decimal expansion are 527. What is the smallest possible value of $a+b?$ | 55 |
If we divide the number $19250$ by a certain number, the remainder is $11$. Similarly, if we divide the number $20302$ by the same number, the remainder is $3$. What is this number? | 53 |
Let $AB$ be a chord of a circle with center $O$. Let $C$ be a point on the circle such that $\angle ABC = 30^\circ$ and $O$ lies inside the triangle $ABC$. Let $D$ be a point on $AB$ such that $\angle DCO = \angle OCB = 20^\circ$. Find the measure of $\angle CDO$ in degrees. | 80 |
Lucy has $8$ children, each of whom has a distinct favorite integer from $1$ to $10$, inclusive. The smallest number that is a perfect multiple of all of these favorite numbers is $1260$, and the average of these favorite numbers is at most $5$. Find the sum of the four largest numbers. | 27 |
Suppose there are 36 objects placed at equal distances along a circle. Determine the number of ways to choose 3 objects such that no two chosen objects are adjacent or diametrically opposite. | 5412 |
A pen costs \(11\) € and a notebook costs \(13\) €. Find the number of ways in which a person can spend exactly \(1000\) € to buy pens and notebooks. | 7 |
Define $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$. Let the sum of all $H_n$ that are terminating in base 10 be $S$. If $S = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $100m+n$. | 9920 |
Let $S = \{1, 22, 333, \dots , 999999999\}$. For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$? | 14 |
If we define $\otimes(a,b,c)$ by \begin{align*} \otimes(a,b,c) = \frac{\max(a,b,c)- \min(a,b,c)}{a+b+c-\min(a,b,c)-\max(a,b,c)}, \end{align*} compute $\otimes(\otimes(7,1,3),\otimes(-3,-4,2),1)$. | 4 |
In a city, there are $2019$ metro stations. Some pairs of stations are connected by tunnels, and from any station, you can reach any other station through these tunnels. The mayor ordered the organization of several metro lines: each line should include several different stations connected in series by tunnels (several... | 1008 |
What is the $22^{\text{nd}}$ positive integer $n$ such that $22^n$ ends in a $2$ (when written in base $10$)? | 85 |
Determine the last three digits of $2003^{2002^{2001}}$. | 241 |
For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1} = f_1 \circ f_n$. Evaluate $f_{2007}(2^{2006})$. | 169 |
An elephant writes a sequence of numbers on a board starting with $1$. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number o... | 2 |
Add one pair of brackets to the expression: \[1 + 2 \times 3 + 4 \times 5 + 6\] so that the resulting expression has a valid mathematical value. For example, \(1 + 2 \times (3 + 4 \times 5) + 6 = 53\). What is the largest possible value that one can make? | 77 |
Determine the smallest positive integer containing only $0$ and $1$ as digits that is divisible by each integer from $1$ through $9$. | 1111111110 |
In how many ways can the number $2000$ be written as a sum of three positive, not necessarily different integers? (Sums like $1 + 2 + 3$ and $3 + 1 + 2$ etc. are considered the same.) | 333333 |
What is the minimum possible perimeter of a right triangle with integer side lengths whose perimeter is equal to its area? | 24 |
Let $ABC$ be a triangle such that $AB = AC$. Suppose the tangent to the circumcircle of $\triangle ABC$ at $B$ is perpendicular to $AC$. Find the measure of angle $\angle ABC$ in degrees. | 30 |
Compute $$ 2022^{\left(2022^{\cdot ^ {\cdot ^{\cdot ^{\left(2022^{2022}\right)}}}}\right)} \pmod{111} $$ where there are $2022$ instances of $2022$ in the tower. (Give the answer as an integer from $0$ to $110$). | 75 |
Suppose that $a$ and $b$ are positive real numbers such that: \[ 3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab). \] Find $101 - \sqrt[3]{a} - \sqrt[3]{b}$. | 0 |
For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which \[r(n, 1000) > r(n, 1001).\] | 499500 |
Let $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, for $i = 1, 2, \dots, m$. It is known that every two elements of $X$ are contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$. | 50 |
Evaluate the expression: \[2 \times (2 \times (2 \times (2 \times (2 \times (2 \times 2 - 2) - 2) - 2) - 2) - 2) - 2.\] | 2 |
\[\left(1 + \frac{1}{1+2^1}\right)\left(1+\frac{1}{1+2^2}\right)\left(1 + \frac{1}{1+2^3}\right)\cdots\left(1 + \frac{1}{1+2^{10}}\right)= \frac{m}{n},\] where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 3073 |
We have a positive integer $n$, whose sum of digits is 100. If the sum of digits of $44n$ is 800, then what is the sum of digits of $3n$? | 300 |
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 |
Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$. | 9 |
Find the number of real numbers $x$ that satisfy the equation $(3^x)^{x+2} + (4^x)^{x+2} - (6^x)^{x+2} = 1$. | 4 |
Let $G=10^{10^{100}}$ (also known as a googolplex). Then, \[ \log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G \] can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine the sum of the digits of $m+n$. | 18 |
Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14th jum... | 613470 |
Given two positive integers $m$ and $n$, we say that $m \mid\mid n$ if $m \mid n$ and $\gcd(m,\, n/m) = 1$. Compute the smallest integer greater than \[ \sum_{d \mid 2016} \sum_{m \mid\mid d} \frac{1}{m}. \] | 52 |
Line $\ell$ passes through point $A$ and into the interior of the equilateral triangle $ABC$. Points $D$ and $E$ are the orthogonal projections of points $B$ and $C$ onto line $\ell$, respectively. Given that $DE = 1$ and $2BD = CE$, determine the area of triangle $ABC$ expressed as $m\sqrt{n}$, where $m$ and $n$ are p... | 10 |
Given a triangle $ABC$ with an area of $1$. Let $A'$, $B'$, and $C'$ be the midpoints of the sides $[BC]$, $[CA]$, and $[AB]$ respectively. What is the minimal possible area of the common part of the two triangles $A'B'C'$ and $KLM$, if the points $K$, $L$, and $M$ are lying on the segments $[AB']$, $[CA']$, and $[BC']... | 0 |
Compute the number of (not necessarily convex) polygons in the coordinate plane with the following properties: - If the coordinates of a vertex are $(x,y)$, then $x,y$ are integers and $1\leq |x|+|y|\leq 3$. - Every side of the polygon is parallel to either the x or y axis. - The point $(0,0)$ is contained in the inter... | 256 |
Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$. Let $c$ be the coefficient of $x^{48}$ in $f(x)$. When $c$ is divided by 101, what is the remainder? | 60 |
Triangle $\triangle ABC$ has $AB = 8$, $AC = 10$, and $AD = \sqrt{33}$, where $D$ is the midpoint of $BC$. Perpendiculars are drawn from $D$ to meet $AB$ and $AC$ at $E$ and $F$, respectively. The length of $EF$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, c$ are relatively prime and $b$ is square-free. Compute... | 33 |
Given a natural number $n$, suppose that $N$ is the maximum number of elephants (bishops) that can be placed on a chessboard measuring $2 \times n$ such that no two elephants are mutually under attack. Determine the number of ways to place $N$ elephants on a $2 \times n$ chessboard so that no two elephants attack each ... | 2 |
There is $1$ integer between $300$ and $400$ (inclusive) in base $10$ such that its last digit is $7$ when written in bases $8$, $10$, and $12$. Find this integer in base $10$. | 367 |
Evaluate \[2 \times (2 \times (2 \times (2 \times (2 \times (2 \times 2 - 2) - 2) - 2) - 2) - 2) - 2.\] | 2 |
For any positive integer $k$, define $f_1(k)$ as the square of the digital sum of $k$ in the decimal system. For $n > 1$, define $f_{n}(k)=f_1(f_{n-1}(k))$. Compute $f_{1992}(2^{1991})$. | 256 |
Points $E$ and $F$ lie inside rectangle $ABCD$ such that $AE = DE = BF = CF = EF$. Given $AB = 11$ and $BC = 8$, find the area of quadrilateral $AEFB$. | 32 |
How many integers $x$, from $10$ to $99$ inclusive, have the property that the remainder of $x^2$ divided by $100$ is equal to the square of the units digit of $x$? | 26 |
Alice, Bob, Charlie, Diana, Emma, and Fred sit in a circle, in that order, and each roll a six-sided die. Each person looks at their own roll and also looks at the roll of either the person to the right or to the left, deciding at random. Then, at the same time, Alice, Bob, Charlie, Diana, Emma, and Fred each state the... | 1440 |
A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, it holds that $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. | 2842 |
Determine the largest three-digit prime number that is a factor of \( \binom{2000}{1000} \). | 661 |
$A, B, C$, and $D$ are points on a circle, forming a convex quadrilateral $ABCD$. Given that $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, find the area of quadrilateral $ABCD$. | 504 |
Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and let $P = AC \cap BD$. Given $AB = 21$, $CD = 7$, $AD = 13$, and the area of trapezoid $[ABCD] = 168$. A line parallel to $AB$ through $P$ intersects the circumcircle of $\triangle BCP$ at $X$. The circumcircles of $\triangle BCP$ and $\triangle APD$ intersect at... | 17 |
If $15$ and $9$ are the lengths of two medians of a triangle, what is the maximum possible area of the triangle, rounded to the nearest integer? | 90 |
What is the smallest positive integer $n$ such that $n = x^3 + y^3$ for two different positive integer tuples $(x, y)$? | 1729 |
The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day, they caught $3$ fewer rabbits and $2$ more squirrels than the day before. This pattern continued, with them catching $3$ fewer rabbits and $2$ more squirrels each day. Determine the total number of rabbi... | 5491 |
In a round-robin chess tournament organized at a school, every two students played one match against each other. Determine the minimum possible number of students in the school given the following conditions: - Each girl student has at least 21 wins in matches against boy students. - Each boy student has at least 12 wi... | 65 |
Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits. | 1944 |
Given the digits \( \{0, 1, 2, \ldots, 9\} \). Find the number of 6-digit numbers that contain the digit \(7\) and can be permuted. For example, 137456 and 314756 are considered the same number. | 2002 |
If $m$ and $n$ are integers such that $3m + 4n = 100$, what is the smallest possible value of $|m - n|$? | 3 |
All possible $6$-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g., $877550$), are written as a sequence in increasing order. Find the $2005^{\text{th}}$ number in this sequence. | 864100 |
For an odd number $n$, we define $n!! = n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1$. Determine how many different residues modulo $1000$ are obtained from $n!!$ when $n = 1, 3, 5, \ldots$. | 15 |
Let $\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$. Let $C'$ denote the reflection of $C$ over the midpoint of $\overline{AX}$. If $BC'=30$, find the sum of all possible side lengths of $\triangle ABC$. | 130 |
A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits? | 53 |
Let $n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}=\prod_{i=1}^k p_i^{e_i}$, where $p_1<p_2<\dots<p_k$ are primes and $e_1, e_2, \dots, e_k$ are positive integers. Let $f(n) = \prod_{i=1}^k e_i^{p_i}$. Find the number of integers $n$ such that $2 \le n \le 2023$ and $f(n)=128$. | 35 |
How many integers $n > 1$ are there such that $n$ divides $x^{13} - x$ for every positive integer $x$? | 31 |
Let $a, b, c, d$ be an increasing arithmetic sequence of positive real numbers with a common difference of $\sqrt{2}$. Given that the product $abcd = 2021$, $d$ can be expressed as $\frac{m+\sqrt{n}}{\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers that are not divisible by the square of any prime. Find $m + n... | 100 |
During a day, 2016 customers visited the store. Each customer visited the store only once (a customer enters the store, spends some time, and then leaves). Find the greatest integer $k$ such that the following statement is always true: We can find $k$ customers such that either all of them have been at the store at the... | 45 |
For each positive integer $m$ and $n$, define the function $f(m, n)$ as follows: - $f(1, 1) = 1$ - $f(m+1, n) = f(m, n) + m$ - $f(m, n+1) = f(m, n) - n$ Find the sum of all values of $p$ such that $f(p, q) = 2004$ for some $q$. | 3007 |
A subset $B$ of \( \{1, 2, \dots, 2017\} \) is said to have property \( T \) if any three elements of \( B \) are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property \( T \) may contain. | 1009 |
Meghana writes two (not necessarily distinct) prime numbers $q$ and $r$ in base 10 next to each other on a blackboard, resulting in the concatenation of $q$ and $r$. For example, if $q = 13$ and $r = 5$, the number on the blackboard is $135$. She notices that three more than the resulting number is the square of a prim... | 5 |
Consider the permutation of $1, 2, \ldots, n$, which we denote as $\{a_1, a_2, \ldots, a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions: 1. $a_1 = 1$ 2. $|a_i - a_{i-1}| \le 2$, for $i = 2, 3, \ldots, n$ What is the residue when we divide $f(2015)$ by $4$? | 3 |
An ant is at a vertex of a cube. Every $10$ minutes, it moves to an adjacent vertex along an edge. If $N$ is the number of one-hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$. | 74 |
Alice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide on a specific time. Each of the four group members signs on to Zoom at a uniformly random time between 1 and 2 PM, and they stay for 1 hour. The group gets work done whenever at least three members are pres... | 48 |
A blind ant is walking on the coordinate plane. It is trying to reach an anthill, placed at all points where both the $x$-coordinate and $y$-coordinate are odd. The ant starts at the origin, and each minute it moves one unit either up, down, to the right, or to the left, each with probability $\frac{1}{4}$. The ant mov... | 3 |
Circle $\Gamma$ with radius $1$ is centered at point $A$ on the circumference of circle $\omega$ with radius $7$. Suppose that point $P$ lies on $\omega$ with $AP=4$. Determine the product of the distances from $P$ to the two intersections of $\omega$ and $\Gamma$. | 15 |
On Andover's campus, Graves Hall is $60$ meters west of George Washington Hall, and George Washington Hall is $80$ meters north of Paresky Commons. Jessica wants to walk from Graves Hall to Paresky Commons. If she first walks straight from Graves Hall to George Washington Hall and then walks straight from George Washin... | 150 |
A sequence consists of the digits $122333444455555\ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501^{\text{st}}$ and $4052^{\text{nd}}$ digits of this sequence. | 13 |
Evan has $66000$ omons, particles that can cluster into groups of a perfect square number of omons. An omon in a cluster of $n^2$ omons has a potential energy of $\frac{1}{n}$. Evan accurately computes the sum of the potential energies of all the omons. Compute the smallest possible value of his result. | 284 |
On a straight line $\ell$, there are four points, $A$, $B$, $C$, and $D$ in that order, such that $AB = BC = CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed. Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that ... | 30 |
Find the smallest positive integer $n$ such that for any $n$ distinct integers $a_1, a_2, \dots, a_n$, the product of all differences $a_i - a_j$ for $i < j$ is divisible by $1991$. | 182 |
Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x, y)$ such that $1 \leq x, y \leq p$ and $x^3 - 3xy + y^3 + 1$ is divisible by $p$. | 30024 |
Compute the sum of all positive integers $n$ such that the median of the $n$ smallest prime numbers is $n$. | 25 |
Adam has a box with $15$ pool balls numbered from $1$ to $15$. He picks out $5$ of them and sorts them in increasing order. He then calculates the four differences between each pair of adjacent balls and finds that exactly two of these differences are equal to $1$. How many different selections of $5$ balls could he ha... | 990 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.