prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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In Prime Land, there are seven major cities, labeled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e., we take the indices modulo $7$. Al initially starts at city $C_0$. Each minute for ten minutes, Al flips a fair coin. If the coin lands heads, and he is at city $C_k$,... | 147 |
The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order). The common difference of this arithmetic sequence can be any real number in the open interval $(-D, D)$. Compute the greatest possible value of $D$. | 24 |
The function $f : \mathbb{R} \to \mathbb{R}$ satisfies the equation \( f(x^2) f''(x) = f'(x) f'(x^2) \) for all real numbers \( x \). Given that \( f(1) = 1 \) and \( f'''(1) = 8 \), determine the value of \( f'(1) + f''(1) \). | 6 |
The value of $21!$ is $51,090,942,171,abc,440,000$, where $a$, $b$, and $c$ are digits. What is the value of $100a + 10b + c$? | 709 |
A cube $ABCDA'B'C'D'$ is given with an edge of length $2$, with vertices marked as in the figure. The point $K$ is the center of the edge $AB$. The plane containing the points $B', D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$. | 1 |
Let $r$, $s$, and $t$ be the roots of the polynomial $x^3 + 6x^2 + 7x + 8$. Calculate the expression: \[ (r^2 + s + t)(s^2 + t + r)(t^2 + r + s) \] | 1100 |
How many zeros does $101^{100} - 1$ end with? | 4 |
Find the number of integer solutions to $||x| - 2020| < 5$. | 18 |
Find the last eight digits of the binary representation of $27^{1986}$. | 11011001 |
The roots of the equation $x^5 - 180x^4 + Ax^3 + Bx^2 + Cx + D = 0$ are in geometric progression. The sum of their reciprocals is $20$. Compute $|D|$. | 243 |
The figure on the right shows the ellipse \( \frac{(x-19)^2}{19} + \frac{(y-98)^2}{98} = 1998 \). Let \( R_1, R_2, R_3, \) and \( R_4 \) denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of \( R_1 - R_2 + R_3 - R_4 \). | 7448 |
Let $x \neq y$ be positive reals satisfying $x^3 + 2013y = y^3 + 2013x$, and let $M = (\sqrt{3} + 1)x + 2y$. Determine the maximum possible value of $M^2$. | 16104 |
Let $\triangle ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 6$. The internal angle bisector of $\angle C$ intersects the side $AB$ at point $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM \parallel AC$ and $DN \parallel BC$. If $(MN)^2 = \frac{p}{q}$, where $p$ and $q$ are... | 2 |
Kevin is trying to solve an economics question which has six steps. At each step, he has a probability $p$ of making a sign error. Let $q$ be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of $0 \le p \le 1$ is it true that $p+q=1$? | 2 |
Find $ \lim_{x\rightarrow 0} \frac{1}{x} \int_{-x}^x (t\sin 2006 t + 2007t + 1004)\ dt$. | 2008 |
Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the center $O$ lies between these chords. Suppose $AB = 6$ and $CD = 8$. Further, suppose that the area of the part of the circle lying between the chords $AB$ and $CD$ is given by \( \frac{m\pi + n}{k} \), where $m$, $n$, and $k$ are positi... | 75 |
Find the largest prime that divides $1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 44 \cdot 45 \cdot 46$. | 47 |
Find the number of nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number. | 4 |
Positive integers $x, y, z$ satisfy $xy + z = 160$. Compute the smallest possible value of $x + yz$. | 50 |
Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom{9}{2} = 36$ lines. Assume that all $36$ lines drawn are distinct. What is the maximum possible number of points which lie on at... | 462 |
Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. Note that numbers like $012$ and $007$ are not considered $3$-digit numbers. | 400185 |
Find the remainder when $2^{5^9} + 5^{9^2} + 9^{2^5}$ is divided by $11$. | 8 |
The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$. Compute the eighth perfect square in the sequence. | 16384 |
What is the last three digits of the sum \(11! + 12! + 13! + \ldots + 2006!\)? | 400 |
What is the smallest number $k$ greater than $6$ for which: \[1 + 2 + \ldots + k = k + (k+1) + \ldots + n\] where $n$ is an integer greater than $k$? | 35 |
Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible... | 13 |
Let $\omega$ be a circle with diameter $AB$ and center $O$. We draw a circle $\omega_A$ through $O$ and $A$, and another circle $\omega_B$ through $O$ and $B$; the circles $\omega_A$ and $\omega_B$ intersect at a point $C$ distinct from $O$. Assume that all three circles $\omega$, $\omega_A$, $\omega_B$ are congruent. ... | 6 |
A company that sells keychains incurs a daily maintenance cost of $\$500$. Additionally, each worker is paid $\$15$ per hour. Each worker produces 5 keychains per hour, and each keychain is sold for $\$3.10$. What is the minimum number of workers the company must hire to make a profit during an 8-hour workday? | 126 |
Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the botto... | 112 |
How far is it from the point \((9, 17)\) to its reflection across the line \(3x + 4y = 15\)? | 32 |
Find the numerator of \[ \frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011} \] when reduced. | 9091 |
The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x) \cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$... | 100 |
Let $f(n)=\varphi(n^3)^{-1}$, where $\varphi(n)$ denotes the number of positive integers not greater than $n$ that are relatively prime to $n$. Suppose \[ \frac{f(1)+f(3)+f(5)+\dots}{f(2)+f(4)+f(6)+\dots} = \frac{m}{n} \] where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. | 702 |
Let $a$, $b$, and $c$ be integers simultaneously satisfying the equations: \[ 4abc + a + b + c = 2018 \] and \[ ab + bc + ca = -507 \] Find \(|a| + |b| + |c|\). | 46 |
\( \triangle ABC \) has side lengths \( 13 \), \( 14 \), and \( 15 \). Let the feet of the altitudes from \( A \), \( B \), and \( C \) be \( D \), \( E \), and \( F \), respectively. The circumcircle of \( \triangle DEF \) intersects \( AD \), \( BE \), and \( CF \) at \( I \), \( J \), and \( K \) respectively. What ... | 21 |
For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$? | 1680 |
An integer $n \ge 2$ is called *delicious* if there exist $n$ positive integers that sum up to 2014 and have distinct remainders when divided by $n$. Determine the smallest delicious integer. | 4 |
A magician has $300$ cards, each with a unique number from $1$ to $300$. These cards are arranged in a $3 \times 100$ rectangle, with one card per unit square. The cards are placed such that consecutive numbers are in neighboring squares, but the numbers are not visible. The magician then turns over $k$ cards of his ch... | 2 |
A building contractor needs to pay his $108$ workers \(\$200\) each. He is carrying $122$ one hundred dollar bills and $188$ fifty dollar bills. Only $45$ workers get paid with two \(\$100\) bills. Find the number of workers who get paid with four \(\$50\) bills. | 31 |
Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$ | 1847 |
A courtyard has the shape of a parallelogram $ABCD$. At the corners of the courtyard, there stand poles $AA'$, $BB'$, $CC'$, and $DD'$, each of which is perpendicular to the ground. The heights of these poles are $AA' = 68$ centimeters, $BB' = 75$ centimeters, $CC' = 112$ centimeters, and $DD' = 133$ centimeters. Find ... | 14 |
Let $\mathcal{P}$ be a parabola that passes through the points $(0, 0)$ and $(12, 5)$. Suppose that the directrix of $\mathcal{P}$ takes the form $y = b$. Find the minimum possible value of $|b|$. | 4 |
Determine the number of ordered pairs of positive integers $(x, y)$ with $y < x \le 100$ such that $x^2 - y^2$ and $x^3 - y^3$ are relatively prime. (Two numbers are relatively prime if they have no common factor other than $1$.) | 99 |
Find all positive integers $n$ such that both $837 + n$ and $837 - n$ are cubes of positive integers. | 494 |
A positive integer is written on each of the six faces of a cube. For each vertex of the cube, we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces? | 31 |
Find the sum of all positive integers $n$ such that \( \frac{2020}{n^3 + n} \) is an integer. | 13 |
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference? | 672 |
Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play? | 7 |
Compute $ \left\lfloor \frac{2005^3}{2003 \cdot 2004} - \frac{2003^3}{2004 \cdot 2005} \right\rfloor $ | 8 |
Let $n$ be a positive integer. Define $f(n)$ as the probability that if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then \( \gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c)) \). Let $s(n)$ be the sum of the distinct prime divisors of $n$. If \( f(n) < \frac{1}{2018} \), compute the ... | 58 |
Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f,a,b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$ and $a,b \in \mathbb{N}_0$ satisfying the following conditions: 1. $f(1) = 2$ 2. $f(a) + f(b) \leq 2\sqrt{f(a)}$ 3. For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$ Find the... | 10201 |
Let $a, b$ be positive real numbers with $a > b$. Compute the minimum possible value of the expression \[ \frac{a^2b - ab^2 + 8}{ab - b^2}. \] | 6 |
How many ordered triples \((a,b,c)\) of integers with \(1 \leq a \leq b \leq c \leq 60\) satisfy \(a \cdot b = c\)? | 134 |
An alarm clock runs 4 minutes slow every hour. It was set right $3 \frac{1}{2}$ hours ago. Now another clock, which is correct, shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon? | 15 |
Find the value of \((2^{40} + 12^{41} + 23^{42} + 67^{43} + 87^{44})^{45! + 46} \mod 11\). | 5 |
Each of the cells of a $7 \times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the expected number of hidden edges is less than $3$... | 29 |
Let $P(x)$ be the product of all linear polynomials $ax+b$, where $a,b \in \{0,\ldots,2016\}$ and $(a,b) \neq (0,0)$. Let $R(x)$ be the remainder when $P(x)$ is divided by $x^5-1$. Determine the remainder when $R(5)$ is divided by $2017$. | 121 |
Find the largest integer $n$, where $2009^n$ divides $2008^{2009^{2010}} + 2010^{2009^{2008}}$. | 2009 |
Let triangle $\triangle ABC$ have side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Let $M$ be the midpoint of $\overline{BC}$, and $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle AEM = m\angle AFM = 90^{\circ}$. Let ... | 28 |
A total of $2010$ coins are distributed in $5$ boxes. Initially, the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that, she can perform the following transformation any number of times: from a box with at least $4$ coins, she can transfer o... | 2004 |
The number $27,000,001$ has exactly four prime factors. Find the sum of these prime factors. | 652 |
In triangle $ABC$, where $\angle ABC$ is obtuse, point $D$ is located on side $AC$ such that $\angle ABD$ is a right angle. Point $E$ is on side $AC$ between points $A$ and $D$, such that line $BD$ bisects $\angle EBC$. Given that $AC = 35$, $BC = 7$, and $BE = 5$, find the length of $CE$. | 10 |
What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$? | 17 |
Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions: 1. Each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are not relatively prime. 2. All other pairs of the five integers are relatively prime. | 47 |
A classroom consists of a $5 \times 5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at ea... | 962 |
The following diagram shows a square where each side has seven dots that divide the side into six equal segments. All the line segments that connect these dots and form a $45^{\circ}$ angle with a side of the square are drawn as shown. The area of the shaded region is 75. Find the area of the original square. | 180 |
Compute the number of non-empty subsets $S$ of $\{-3, -2, -1, 0, 1, 2, 3\}$ with the following property: for any $k \ge 1$ distinct elements $a_1, \dots, a_k \in S$, we have $a_1 + \dots + a_k \neq 0$. | 24 |
The four-digit base ten number $\underline{a}\;\underline{b}\;\underline{c}\;\underline{d}$ has all nonzero digits and is a multiple of $99$. Additionally, the two-digit base ten number $\underline{a}\;\underline{b}$ is a divisor of $150$, and the two-digit base ten number $\underline{c}\;\underline{d}$ is a divisor of... | 108 |
We define the ridiculous numbers recursively as follows: \begin{itemize} \item 1 is a ridiculous number. \item If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers. \end{itemize} A closed interval $I$ is "boring" if: \begin{itemize} \item $I$ contains no ridiculous numbers, and \i... | 9 |
Find all positive integers $n$ such that \( \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2 \), where \( \lfloor x \rfloor \) represents the largest integer less than the real number $x$. | 24 |
How many ordered pairs of integers \((m,n)\) are there such that \(m\) and \(n\) are the legs of a right triangle with an area equal to a prime number not exceeding 80? | 87 |
Let $a_1, a_2, \ldots$ be a sequence defined by $a_1 = 1$ and for $n \ge 1$, $a_{n+1} = \sqrt{a_n^2 - 2a_n + 3} + 1$. Find $a_{513}$. | 33 |
Let $m$ and $n$ denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum $m + n$? | 2008 |
Evaluate the integral: \[ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{2\sin^3 x}{\cos^5 x} \, dx. \] | 4 |
Let $A = \{m : m \text{ is an integer and the roots of } x^2 + mx + 2020 = 0 \text{ are positive integers}\}$ and $B= \{n : n \text{ is an integer and the roots of } x^2 + 2020x + n = 0 \text{ are negative integers}\}$. Suppose $a$ is the largest element of $A$ and $b$ is the smallest element of $B$. Find the sum of th... | 26 |
Find the remainder when $1! + 2! + 3! + \dots + 1000!$ is divided by $9$. | 0 |
In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections. | 222 |
Determine all $n \geq 3$ for which there exist $n$ positive integers $a_1, a_2, \ldots, a_n$ such that any two of them have a common divisor greater than 1, but any three of them are coprime. Furthermore, assuming that the numbers $a_i$ are less than 5000, find the greatest possible $n$. | 4 |
Given that $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b, c, d, e$ are equal to the same integer, solve the following: - $b \leq d$ - $c \geq a$ - $a \leq e$ - $b \geq e$ - $d \neq 5$ Determine the value of $a^b + c^d + e$. | 628 |
Find the remainder when $19^{92}$ is divided by 92. | 49 |
The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became indepe... | 50 |
Navi and Ozna are playing a game where Ozna starts first and the two take turns making moves. A positive integer is written on the board. A move consists of either: 1. Subtracting any positive integer at most $2015$ from it, or 2. If the integer on the board is divisible by $2014$, dividing by $2014$. The first person ... | 2013 |
Suppose $ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB = 10$ and $CD = 15$, compute the maximum possible value of $XY$. | 6 |
In $\triangle ABC$, $\angle A = 30^{\circ}$, $BC = 13$. Given two circles $\gamma_1$ and $\gamma_2$ with radii $r_1$ and $r_2$, respectively, contain point $A$ and touch line $BC$ at points $B$ and $C$. Find the product $r_1r_2$. | 169 |
What is the least positive integer such that the product of its digits is $9!$? | 2578899 |
A week ago, Sandy's seasonal Little League batting average was $0.360$. After five more at bats this week, Sandy's batting average is now $0.400$. What is the smallest number of hits that Sandy could have had this season? | 12 |
Convex quadrilateral $ABCD$ has $AB = 20$, $BC = CD = 26$, and $\angle ABC = 90^\circ$. Point $P$ is on $DA$ such that $\angle PBA = \angle ADB$. If $PB = 20$, compute the area of $ABCD$. | 630 |
Given a square $ABCD$. Let $P \in AB$, $Q \in BC$, $R \in CD$, $S \in DA$ such that $PR \parallel BC$ and $SQ \parallel AB$. Let $Z = PR \cap SQ$. If $BP = 7$, $BQ = 6$, and $DZ = 5$, find the side length of the square. | 10 |
Ann and Drew have purchased a mysterious slot machine. Each time it is spun, it chooses a random positive integer such that $k$ is chosen with probability $2^{-k}$ for every positive integer $k$, and then it outputs $k$ tokens. Let $N$ be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann goin... | 5 |
Construction Mayhem University is expanding its campus with a new construction schedule. A new project begins every two days, with each project taking exactly one more day than the previous one to complete. The first project takes 3 days, the second takes 4 days, and so on. Suppose the new schedule starts on Day 1. On ... | 51 |
Suppose $x, y \in \mathbb{Z}$ satisfy the equation: \[ y^4 + 4y^3 + 28y + 8x^3 + 6y^2 + 32x + 1 = (x^2 - y^2)(x^2 + y^2 + 24). \] Find the sum of all possible values of $|xy|$. | 18 |
How many positive integers less than $200$ are relatively prime to either $15$ or $24$? | 120 |
Four people were guessing the number, $N$, of jellybeans in a jar. No two guesses were equally close to $N$. The closest guess was 80 jellybeans, the next closest guess was 60 jellybeans, followed by 49 jellybeans, and the furthest guess was 125 jellybeans. Find the sum of all possible values for $N$. | 1256 |
Pictured below is part of a large circle with radius $30$. There is a chain of three circles with radius $3$, each internally tangent to the large circle and each tangent to its neighbors in the chain. There are two circles with radius $2$, each tangent to two of the radius $3$ circles. The distance between the centers... | 46 |
On March $6$, $2002$, during the celebrations of the $500$th anniversary of the birth of mathematician Pedro Nunes, ten people entered the Viva bookstore for science. Each person bought exactly $3$ different books. Furthermore, any two people bought at least one copy of the same book. "The Adventures of Mathematics" by... | 5 |
Let $P$ be the product of all the entries in row 2021 of Pascal's triangle (the row that begins 1, 2021, \ldots). What is the largest integer $j$ such that $P$ is divisible by $101^j$? | 1980 |
How many sequences of $5$ positive integers $(a, b, c, d, e)$ satisfy the inequality $abcde \leq a + b + c + d + e \leq 10$? | 116 |
Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that \(\text{area} (\triangle ADE) = 16\), \(\text{area} (\triangle CEF) = 9\) and \(\text{area} (\triangle ABF) = 25\). What is the area of \(\triangle AEF\)? | 30 |
Natural number $M$ has 6 divisors such that the sum of them is equal to 3500. Find all values of $M$. | 1996 |
Let $\ell$ be a real number satisfying the equation $\frac{(1+\ell)^2}{1+\ell^2}=\frac{13}{37}$. Then \[\frac{(1+\ell)^3}{1+\ell^3}=\frac{m}{n},\] where $m$ and $n$ are positive coprime integers. Find $m+n$. | 62 |
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