prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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In trapezoid $ABCD$, where $BC \parallel AD$, $AB = 13$, $BC = 15$, $CD = 14$, and $DA = 30$. Find the area of $ABCD$. | 252 |
What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square? | 7 |
For every positive integer $n$, define $S_n$ to be the sum \[ S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \pi}{2010} \right)^n . \] As $n$ approaches infinity, what value does $S_n$ approach? | 1944 |
Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. What is the minimum number of distinct problems needed? | 13 |
There are $2014$ students from high schools nationwide sitting around a round table in an arbitrary manner. The organizers want to rearrange the students so that those from the same school sit next to each other. This is done by performing swaps between two adjacent groups of students. The task is to find the smallest ... | 2013 |
How many solutions does the equation: $$\left\lfloor \frac{x}{20} \right\rfloor = \left\lfloor \frac{x}{17} \right\rfloor$$ have over the set of positive integers? $\left\lfloor a \right\rfloor$ denotes the largest integer that is less than or equal to $a$. | 56 |
How many cubic polynomials of the form $x^3 - ax^2 + (a+d)x - (a+2d)$ have roots that are all non-negative integers, where $a$ and $d$ are integers? | 3 |
Call a lattice point *visible* if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$. Let $D$ denote the set of all positive divisors of $2021 \cdot 202... | 20 |
How many seven-digit numbers are multiples of $388$ and end in $388$? | 93 |
Let $S$ be a subset of \( \{1, 2, \ldots, 500\} \) such that no two distinct elements of $S$ have a product that is a perfect square. Find, with proof, the maximum possible number of elements in $S$. | 306 |
Let $A$ be a positive integer which is a multiple of 3 but not a multiple of 9. If adding the product of each digit of $A$ to $A$ results in a multiple of 9, find the possible minimum value of $A$. | 138 |
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive integer that is the product of three distinct prime numbers!" Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, ... | 70 |
The area of triangle $ABC$ is $2007$. One of its sides has length $18$, and the tangent of the angle opposite that side is $\frac{2007}{24832}$. When the altitude is dropped to the side of length $18$, it cuts that side into two segments. Find the sum of the squares of those two segments. | 194 |
Determine all positive integers $n$ for which $2^{n-1}n + 1$ is a perfect square. | 5 |
Students are in a classroom with $n$ rows. In each row, there are $m$ tables. It is given that $m, n \geq 3$. At each table, there is exactly one student. We call the neighbors of a student those sitting one place to the right, left, in front, and behind him. Each student shook hands with his neighbors. In the end, the... | 153 |
Let $AX$ be a diameter of a circle $\Omega$ with radius $10$, and suppose that $C$ lies on $\Omega$ so that $AC = 16$. Let $D$ be the other point on $\Omega$ so that $CX = CD$. From here, define $D'$ to be the reflection of $D$ across the midpoint of $AC$, and $X'$ to be the reflection of $X$ across the midpoint of $CD... | 1369 |
Find the integer part of \( \sum_{n=1}^{40000} \frac{1}{\sqrt{n}} \). | 398 |
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N}$, including $... | 4 |
Let \( a^3 - a - 1 = 0 \). Find the exact value of the expression \[ \sqrt[3]{3a^2-4a} + a\sqrt[4]{2a^2+3a+2}. \] | 2 |
For every $a \in \mathbb{N}$, denote by $M(a)$ the number of elements in the set \[ \{ b \in \mathbb{N} \mid a + b \text{ is a divisor of } ab \}. \] Find $\max_{a \leq 1983} M(a).$ | 121 |
Find the total number of different integer values the function \[ f(x) = \lfloor x \rfloor + \lfloor 2x \rfloor + \left\lfloor \frac{5x}{3} \right\rfloor + \lfloor 3x \rfloor + \lfloor 4x \rfloor \] takes for real numbers $x$ with $0 \leq x \leq 100$. | 734 |
I have a $2 \times 4$ grid of squares. How many ways can I shade at least one of the squares such that no two shaded squares share an edge? | 40 |
How many positive integers $n$ satisfy the inequality \[\left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \ ?\] Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$. | 15049 |
Given that $0.3010 < \log 2 < 0.3011$ and $0.4771 < \log 3 < 0.4772$, find the leftmost digit of $12^{37}$. | 8 |
What is the greatest common divisor of the set of numbers $\{16^n + 10n - 1 \mid n=1,2,\ldots \}$? | 25 |
A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website? | 1000 |
Harry, Hermione, and Ron go to Diagon Alley to buy chocolate frogs. If Harry and Hermione spent one-fourth of their own money, they would spend $3$ galleons in total. If Harry and Ron spent one-fifth of their own money, they would spend $24$ galleons in total. Everyone has a whole number of galleons, and the number of ... | 6 |
It is given that the number $4^{11} + 1$ is divisible by some prime greater than $1000$. Determine this prime. | 2113 |
In triangle $ABC$, the lengths of all sides are integers, $\angle B = 2 \angle A$ and $\angle C > 90^\circ$. Find the smallest possible perimeter of this triangle. | 77 |
For a positive integer $n$, let $s(n)$ be the sum of the digits of $n$ when $n$ is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6$. Find the sum of the two solutions to the equation $n - 3s(n) = 2022$. | 4107 |
What is the size of the largest subset $S'$ of $S = \{2^x 3^y 5^z : 0 \le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p \mid q$? | 19 |
Let $f_1(n)$ be the number of divisors of $n$, and define $f_k(n) = f_1(f_{k-1}(n))$. Compute the smallest integer $k$ such that $f_k(2013^{2013}) = 2$. | 4 |
A polynomial whose roots are all equal to each other is called a unicorn. Compute the number of distinct ordered triples $(M, P, G)$, where $M$, $P$, $G$ are complex numbers, such that the polynomials $z^3 + M z^2 + Pz + G$ and $z^3 + G z^2 + Pz + M$ are both unicorns. | 5 |
Consider a triangle $ABC$ with $AB=AC$, and $D$ as the foot of the altitude from the vertex $A$. The point $E$ lies on the side $AB$ such that $\angle ACE = \angle ECB = 18^{\circ}$. If $AD = 3$, find the length of the segment $CE$. | 6 |
Determine the values of the real parameter $a$, such that the equation \[ \sin 2x \sin 4x - \sin x \sin 3x = a \] has a unique solution in the interval $[0, \pi)$. | 1 |
Find all three-digit numbers $\overline{abc}$ ($a,b \neq 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$. | 627 |
There exist complex numbers $z_1, z_2, \dots, z_{10}$ which satisfy the condition: $$|z_k i^k + z_{k+1} i^{k+1}| = |z_{k+1} i^k + z_k i^{k+1}|$$ for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. Given that $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the min... | 103 |
A permutation \((a_1, a_2, a_3, \dots, a_{2012})\) of \((1, 2, 3, \dots, 2012)\) is selected at random. If \(S\) is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i | \], then compute the sum of the prime factors of \(S\). | 2083 |
For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is: \[ (-1)^{\alpha... | 208 |
Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only... | 382 |
Naomi has a class of $100$ students who will compete with each other in five teams. Once the teams are made, each student will shake hands with every other student, except the students in his or her own team. Naomi chooses to partition the students into teams so as to maximize the number of handshakes. How many handsha... | 4000 |
Find a positive integer $n$ such that if you place the digit $2$ on the left and the digit $1$ on the right of $n$, the resulting number is equal to $33n$. | 87 |
When Applejack begins to buck trees, she starts off with 100 energy. Every minute, she may either choose to buck $n$ trees and lose 1 energy, where $n$ is her current energy, or rest (i.e., buck 0 trees) and gain 1 energy. What is the maximum number of trees she can buck after 60 minutes have passed? | 4293 |
Say that a function $f : \{1, 2, \ldots, 1001\} \to \mathbb{Z}$ is *almost* polynomial if there exists a polynomial $p(x) = a_{200}x^{200} + \ldots + a_1x + a_0$ where each $a_n$ is an integer with $|a_n| \le 201$, and $|f(x) - p(x)| \le 1$ for all $x \in \{1, 2, \ldots, 1001\}$. Let $N$ be the number of such almost po... | 0 |
A lattice point in the plane is a point whose coordinates are both integers. Given a set of 100 distinct lattice points in the plane, find the smallest number of line segments $\overline{AB}$ for which $A$ and $B$ are distinct lattice points in this set, and the midpoint of $\overline{AB}$ is also a lattice point (not ... | 1200 |
Determine the largest integer $n$ such that $7^{2048} - 1$ is divisible by $2^n$. | 14 |
Evaluate the expression: \[2^{7} \times 3^{0} + 2^{6} \times 3^{1} + 2^{5} \times 3^{2} + \cdots + 2^{0} \times 3^{7}.\] | 6305 |
Find the coefficient of $x^{7}y^{6}$ in $(xy + x + 3y + 3)^{8}$. | 672 |
Compute \( \gcd(5^{2547} - 1, 5^{2004} - 1) \). | 124 |
A lucky number is a number whose digits are only $4$ or $7$. What is the $17^{th}$ smallest lucky number? | 4474 |
Natural numbers $k, l, p$, and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$, then $a + \frac{1}{b}$ and $b + \frac{1}{a}$ are the roots of $x^2 - px + q = 0$. What is the sum of all possible values of $q$? | 4 |
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, and $97$. Suppose the curve $\mathcal{V}$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal{V}$ at each of $A$, $B$, and $C$. | 237 |
Maria has $14$ days to train for an olympiad. The conditions are: 1. She cannot train for $3$ consecutive days. 2. She cannot rest for $3$ consecutive days. Determine how many configurations of training days she can achieve her goal. | 1220 |
The square $BCDE$ is inscribed in circle $\omega$ with center $O$. Point $A$ is the reflection of $O$ over $B$. A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$). Assume $BCDE$ has area $200$. To the nearest integer, what is the length of the hook? | 57 |
A point is chosen uniformly at random from the interior of a unit square. Let $p$ be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that $p^2$ can be written as $m-\sqrt{n}$ for positive integers $m$ and $n$, what is $m+n$? | 11 |
How many subsets of the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \) have the property that no two of its elements differ by more than 5? For example, count the sets \( \{3\} \), \( \{2, 5, 7\} \), and \( \{5, 6, 7, 8, 9\} \) but not the set \( \{1, 3, 5, 7\} \). | 256 |
Compute the remainder when $2^{3^5} + 3^{5^2} + 5^{2^3}$ is divided by $30$. | 6 |
On the sides $BC$ and $CD$ of the square $ABCD$ with side length $1$, points $E$ and $F$ are chosen such that $\angle EAB = 20^\circ$ and $\angle EAF = 45^\circ$. Calculate the distance from point $A$ to the line $EF$. | 1 |
Yasha writes in the cells of a $99 \times 99$ table all positive integers from 1 to $99^2$, with each number appearing once. Grisha looks at the table and selects several cells such that no two selected cells share a common side. Grisha then sums up the numbers in all the selected cells. Determine the largest sum that ... | 24017351 |
Misha has accepted a job in the mines and will produce one ore each day. At the market, he can perform the following transactions: - Buy or sell one ore for $3. - Buy or sell bundles of three wheat for $12 each. - Sell one wheat for one ore. His ultimate goal is to build a city, which requires: - Three ore - Two wheat ... | 9 |
The 16 fields of a $4 \times 4$ checkerboard can be arranged in 18 lines as follows: the four rows, the four columns, the five diagonals from northwest to southeast, and the five diagonals from northeast to southwest. These diagonals consist of 2, 3, or 4 edge-adjacent fields of the same color; the corner fields of the... | 16 |
Find the number of arithmetic sequences $a_1, a_2, a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence. | 4 |
Equilateral triangle $ABC$ has a side length of $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and the minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the "lune" that is inside the semicircle but outside sector $ABC$ can be expressed in the form... | 10 |
The diagram below shows the regular hexagon $BCEGHJ$ surrounded by the rectangle $ADFI$. Let $\theta$ be the measure of the acute angle between the side $\overline{EG}$ of the hexagon and the diagonal of the rectangle $\overline{AF}$. There are relatively prime positive integers $m$ and $n$ such that $\sin^2\theta = \f... | 55 |
How many ordered pairs of integers \((x, y)\) satisfy the equation \(x^2 + y^2 + xy = 4(x + y)\)? | 3 |
$A = 1 \cdot 4 \cdot 7 \cdots 2014$. Find the last non-zero digit of $A$ if it is known that $A \equiv 1 \pmod{3}$. | 2 |
The subnumbers of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37 are pr... | 9 |
Let $ABC$ be an equilateral triangle with an area of $1998\ \text{cm}^2$. Points $K$, $L$, and $M$ divide the segments $[AB]$, $[BC]$, and $[CA]$, respectively, in the ratio $3:4$. The line $AL$ intersects the lines $CK$ and $BM$ at points $P$ and $Q$, respectively. The line $BM$ intersects the line $CK$ at point $R$. ... | 54 |
Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has. | 8 |
Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$ | 4620 |
Let $ABCD$ and $AEFG$ be unit squares such that the area of their intersection is $\frac{20}{21}$. Given that $\angle BAE < 45^\circ$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$. | 4940 |
Q. A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along with its interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is ... | 21 |
Rectangle $ABCD$ measures $70$ by $40$. Eighteen points (including $A$ and $C$) are marked on the diagonal $AC$, dividing the diagonal into $17$ congruent pieces. Twenty-two points (including $A$ and $B$) are marked on the side $AB$, dividing the side into $21$ congruent pieces. Seventeen non-overlapping triangles are ... | 600 |
Find the maximum possible value obtainable by inserting a single set of parentheses into the expression $1 + 2 \times 3 + 4 \times 5 + 6$. | 77 |
Let $(x, y)$ be an intersection of the equations $y = 4x^2 - 28x + 41$ and $x^2 + 25y^2 - 7x + 100y + \frac{349}{4} = 0$. Find the sum of all possible values of $x$. | 14 |
When Eva counts, she skips all numbers containing a digit divisible by 3. For example, the first ten numbers she counts are 1, 2, 4, 5, 7, 8, 11, 12, 14, 15. What is the $100^{\text{th}}$ number she counts? | 255 |
Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$? | 444 |
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to form such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of... | 62 |
Let $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given that $RI=3$, $CE=8$, and $ER=7$, compute $IC$. | 4 |
Let $x_1, x_2, \dots, x_{100}$ be real numbers such that $|x_1| = 63$ and $|x_{n+1}| = |x_n + 1|$ for $n = 1, 2, \dots, 99$. Find the largest possible value of $(-x_1 - x_2 - \cdots - x_{100})$. | 2034 |
Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different suits. At the beginning, they are given $20$ cards each. Each turn, one player can either place a card on the table or remove cards from the table whose sum is $15$. At the end of the game: - One player has a $5$ and a... | 8 |
Let $T = (a,b,c)$ be a triangle with sides $a$, $b$, and $c$, and area $\triangle$. Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a$, $b' = h_b$, $c' = h_c$) and denote its area by $\triangle'$. Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes... | 45 |
In the quadrilateral $ABCD$, angles $\angle B$ and $\angle C$ are equal to $120^\circ$. Given $AB = CD = 1$ and $CB = 4$, find the length of $AD$. | 5 |
A $6$-inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches. | 12 |
How many ordered triples of nonzero integers $(a, b, c)$ satisfy the equation $2abc = a + b + c + 4$? | 6 |
Given the list \[A=[9,12,1,20,17,4,10,7,15,8,13,14],\] we would like to sort it in increasing order. To accomplish this, we will perform the following operation repeatedly: remove an element, then insert it at any position in the list, shifting elements if necessary. What is the minimum number of applications of this o... | 6 |
Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other i... | 13 |
Let $A, B, M, C, D$ be distinct points on a line such that $AB = BM = MC = CD = 6$. Circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ and radii $4$ and $9$ are tangent to line $AD$ at $A$ and $D$ respectively, such that $O_1$ and $O_2$ lie on the same side of line $AD$. Let $P$ be the point such that $PB \... | 65 |
A geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd positive integer. How many possible values of $n$ are there? | 8 |
Suppose \( \{a_1,a_2,\ldots,a_7\} \) is a set of pair-wisely different positive integers. If \( a_1, 2a_2, \ldots, 7a_7 \) can form an arithmetic series (in this order), find the smallest positive value of \(|a_7-a_1|\). | 360 |
Let $S = 2^3 + 3^4 + 5^4 + 7^4 + \cdots + 17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$. | 93 |
An integer $n \geq 2$ is said to be *tuanis* if, when you add the smallest prime divisor of $n$ and the largest prime divisor of $n$ (these divisors can be the same), you obtain an odd result. Calculate the sum of all *tuanis* numbers that are less than or equal to $2023$. | 1021086 |
Bob chooses a $4$-digit binary string uniformly at random and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob's string is $0000$ and the str... | 19 |
Let $w, x, y,$ and $z$ be nonzero complex numbers, and let $n$ be a positive integer. Suppose that the following conditions hold: - $\frac{1}{w} + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3,$ - $wx + wy + wz + xy + xz + yz = 14,$ - $(w + x)^3 + (w + y)^3 + (w + z)^3 + (x + y)^3 + (x + z)^3 + (y + z)^3 = 2160,$ - $w + ... | 47 |
There are seven cards in a hat, and on card $k$ there is a number $2^{k-1}$, where $k=1,2,...,7$. Solarin picks the cards at random from the hat, one card at a time, until the sum of the numbers on the cards in his hand exceeds $124$. What is the most probable sum he can get? | 127 |
If $f$ is a monic cubic polynomial with $f(0) = -64$, and all roots of $f$ are non-negative real numbers, what is the largest possible value of $f(-1)$? (Note: A polynomial is monic if it has a leading coefficient of $1$.) | -125 |
A rectangular grid with side lengths that are integers greater than $1$ is given. Smaller rectangles, each with an area equal to an odd integer and each side length also an integer greater than $1$, are cut out one by one. Finally, one single unit is left. Find the least possible area of the initial grid before the cut... | 121 |
In how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ such that $x < y < z$? | 784 |
How many positive integers $N$ satisfy all of the following three conditions? 1. $N$ is divisible by $2020$. 2. $N$ has at most $2020$ decimal digits. 3. The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros. | 508536 |
Temerant is a spherical planet with a radius of $1000$ kilometers. The government wants to build twelve towers of the same height on the equator of Temerant, so that every point on the equator can be seen from at least one tower. The minimum possible height of the towers can be written, in kilometers, as $a\sqrt{b} - c... | 3008 |
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