prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Let $x$, $y$, and $z$ be consecutive integers such that \[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} > \frac{1}{45}. \] Find the maximum value of $x + y + z$. | 405 |
A circumference was divided into $n$ equal parts. On each of these parts, a number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$, and $17$ were in consecutive positions. In how many parts was the circumference divided? | 20 |
In the following diagram, a semicircle is folded along a chord $AN$ and intersects its diameter $MN$ at $B$. Given that $MB : BN = 2 : 3$ and $MN = 10$. If $AN = x$, find $x^2$. | 80 |
At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem, you can earn an integer score from 0 to 7. The contestant's score is the product of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks... | 1 |
Jar #1 contains five red marbles, three blue marbles, and one green marble. Jar #2 contains five blue marbles, three green marbles, and one red marble. Jar #3 contains five green marbles, three red marbles, and one blue marble. You randomly select one marble from each jar. Given that you select one marble of each color... | 323 |
Let $a_d$ be the number of non-negative integer solutions $(a, b)$ to $a + b = d$ where $a \equiv b \pmod{n}$ for a fixed $n \in \mathbb{Z}^+$. Consider the generating function $M(t) = a_0 + a_1t + a_2t^2 + \ldots$ Consider $$P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right).$$ Then $P(n)$, $n \in \mathbb... | 0 |
A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$. | 100 |
A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, t... | 3003 |
Find the number of degree $8$ polynomials $f(x)$ with nonnegative integer coefficients satisfying both $f(1) = 16$ and $f(-1) = 8$. | 47775 |
Find the number of three-digit palindromes that are divisible by $3$. Recall that a palindrome is a number that reads the same forward and backward, such as $727$ or $905509$. | 30 |
Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of ... | 9799 |
Calculate the sum $\sum_{n=1}^{1,000,000} \lfloor \sqrt{n} \rfloor$. You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof. | 666167500 |
Let $\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x \le 100$ and $\text{lcm}(16,x) = 16x$. | 2500 |
Determine the sum of all positive integers $n$ between $1$ and $100$ inclusive such that \(\gcd(n,2^n - 1) = 3.\) | 360 |
Compute the positive integer $n$ such that $\log_3 n < \log_2 3 < \log_3 (n + 1)$. | 5 |
Find the sum of all prime numbers that can be expressed both as a sum of two prime numbers and as a difference of two prime numbers. | 5 |
The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$. Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$, where $m$, $n$ are relatively prime integers. Find $m + n$. | 2224 |
Rectangle $ABCD$ has sides $AB = 10$ and $AD = 7$. Point $G$ lies in the interior of $ABCD$, a distance $2$ from side $\overline{CD}$ and a distance $2$ from side $\overline{BC}$. Points $H, I, J$, and $K$ are located on sides $\overline{BC}, \overline{AB}, \overline{AD}$, and $\overline{CD}$, respectively, so that the... | 8 |
Compute the smallest positive integer $n$ for which $\sqrt{100+\sqrt{n}} + \sqrt{100-\sqrt{n}}$ is an integer. | 6156 |
Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\frac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability... | 743 |
At a concert, $10$ singers will perform. For each singer $x$, either there is a singer $y$ such that $x$ wishes to perform right after $y$, or $x$ has no preferences at all. Suppose that there are $n$ ways to order the singers such that no singer has an unsatisfied preference, and let $p$ be the product of all possible... | 38 |
$P(x)$ is a polynomial of degree $3n$ such that: \[ \begin{align*} P(0) = P(3) = \cdots &= P(3n) = 2, \\ P(1) = P(4) = \cdots &= P(3n-2) = 1, \\ P(2) = P(5) = \cdots &= P(3n-1) = 0, \\ \text{and } & P(3n+1) = 730. \end{align*} \] Determine $n$. | 4 |
Let $C$ be the number of ways to arrange the letters of the word CATALYSIS, $T$ be the number of ways to arrange the letters of the word TRANSPORT, $S$ be the number of ways to arrange the letters of the word STRUCTURE, and $M$ be the number of ways to arrange the letters of the word MOTION. What is $\frac{C - T + S}{M... | 126 |
Let $X = \{1, 2, \ldots, 2017\}$. Let $k$ be a positive integer. Given any $r$ such that $1 \le r \le k$, there exist $k$ subsets of $X$ such that the union of any $r$ of them is equal to $X$, but the union of any fewer than $r$ of them is not equal to $X$. Find, with proof, the greatest possible value for $k$. | 2017 |
Let $z_1$ and $z_2$ be the zeros of the polynomial $f(x) = x^2 + 6x + 11$. Compute $(1 + z_1^2 z_2)(1 + z_1 z_2^2)$. | 1266 |
At a meeting of $12k$ people, each person exchanges greetings with exactly $3k + 6$ others. For any two people, the number of people who exchange greetings with both is the same. How many people are at the meeting? | 36 |
For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$. | 161 |
Reimu invented a new number base system that uses exactly five digits. The number $0$ in the decimal system is represented as $00000$. Whenever a number is incremented, Reimu finds the leftmost digit (of the five digits) that is equal to the "units" (rightmost) digit, increments this digit, and sets all the digits to i... | 1727 |
Let $A, B, C, D$ be points, in order, on a straight line such that $AB = BC = CD$. Let $E$ be a point closer to $B$ than $D$ such that $BE = EC = CD$ and let $F$ be the midpoint of $DE$. Let $AF$ intersect $EC$ at $G$ and let $BF$ intersect $EC$ at $H$. If $[BHC] + [GHF] = 1$, then $AD^2 = \frac{a\sqrt{b}}{c}$ where $a... | 374 |
In the diagram below, $\angle B = 43^\circ$ and $\angle D = 102^\circ$. Find $\angle A + \angle B + \angle C + \angle D + \angle E + \angle F$. | 720 |
Noew is writing a $15$-problem mock AIME consisting of four subjects: algebra, geometry, combinatorics, and number theory. The AIME is considered somewhat evenly distributed if there is at least one problem of each subject and at least six combinatorics problems. Two AIMEs are considered similar if they have the same s... | 84 |
A trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\frac{1}{3}$, and the slope of line $AB$ is $-\frac{2}{3}$. Given that $AB = CD$ and $BC < AD$, the absolute value of the slope of line $CD$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are two relatively prime positive integ... | 1706 |
Compute \[\sum_{n_{60}=0}^2\sum_{n_{59}=0}^{n_{60}}\cdots\sum_{n_2=0}^{n_3}\sum_{n_1=0}^{n_2}\sum_{n_0=0}^{n_1}1.\] | 1953 |
Let $a$ be the sum of the numbers: $$99 \times 0.9$$ $$999 \times 0.9$$ $$9999 \times 0.9$$ $$\vdots$$ $$999\cdots 9 \times 0.9$$ where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$. Find the sum of the digits in the number $a$. | 891 |
The $260$ volumes of the *Encyclopedia Galactica* are out of order in the library. Fortunately for the librarian, the books are numbered. Due to his religion, which holds both encyclopedias and the concept of parity in high esteem, the librarian can only rearrange the books two at a time, and then only by switching the... | 130 |
All of the roots of $x^3 + ax^2 + bx + c$ are positive integers greater than 2, and the coefficients satisfy $a + b + c + 1 = -2009$. Find $a$. | -58 |
Find the sum of all natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$. | 85783 |
Find the largest divisor of $2800$ that is a perfect square. | 400 |
At a meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each other, among the remaining people, they do not have a common friend. For each pair of individuals $A$ and $B$ who are not familiar with each other, there are six common acquaintances among the remaining people. How many people we... | 100 |
Let $f(X, Y, Z) = X^5Y - XY^5 + Y^5Z - YZ^5 + Z^5X - ZX^5$. Find \( \frac{f(2009, 2010, 2011) + f(2010, 2011, 2009) - f(2011, 2010, 2009)}{f(2009, 2010, 2011)} \) | 3 |
A positive integer is called *cool* if it can be expressed in the form $a! \cdot b! + 315$ where $a$ and $b$ are positive integers. For example, $1! \cdot 1! + 315 = 316$ is a cool number. Find the sum of all cool numbers that are also prime numbers. | 317 |
A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$. Find $D$. | 4 |
We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number? | 606 |
Let $\triangle ABC$ be a triangle in a plane such that $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point in three-dimensional space such that $\angle{BDC}=\angle{CDA}=\angle{ADB}=90^\circ$. Let $d$ be the distance from $D$ to the plane containing $\triangle ABC$. The value $d^2$ can be expressed as $\frac{m}{n}$ for re... | 511 |
Consider a rectangle $ABCD$ with $AB = 30$ and $BC = 60$. Construct a circle $T$ whose diameter is $AD$. Construct another circle $S$ whose diameter is $AB$. Let circles $S$ and $T$ intersect at a point $P$ such that $P \neq A$. Let the line $AP$ intersect $BC$ at point $E$. Define point $F$ on $AB$ such that the line ... | 75 |
There are $36$ participants at a BdMO event. Some of the participants shook hands with each other, but no two participants shook hands more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes had shaken hands with each other. ... | 546 |
Which is the largest positive integer $n$ satisfying the inequality: $n^{2007} > (2007)^n$? | 2006 |
Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$. | 70 |
A dartboard consists of three concentric circles with radii 4, 6, and 8. Three darts are thrown at the board, sticking at random locations. Determine the probability that each dart lands in a different region of the dartboard. The probability can be expressed as \( \frac{m}{n} \), where \( m \) and \( n \) are relative... | 617 |
Let $\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC$, $E = BO \cap CA$, $F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{CF}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$. | 43 |
In a right-angled triangle where all side lengths are integers, one of the catheti has a length of $1994$. Determine the length of the hypotenuse. | 994010 |
Determine the last three digits of $374^{2022}$. | 376 |
Determine the largest prime which divides both $2^{24} - 1$ and $2^{16} - 1$. | 17 |
Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^\circ$. Find the measure of $\angle ABC$. | 30 |
Find the sum of the indexes of the singular points other than zero of the vector field: \[ z\overline{z}^2 + z^4 + 2\overline{z}^4 \] | 3 |
What is the remainder when $1234^{567} + 89^{1011}$ is divided by $12$? | 9 |
Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$. | 9 |
Consider an arithmetic progression made up of $100$ terms. If the sum of all the terms of the progression is $150$ and the sum of the even terms is $50$, find the sum of the squares of the $100$ terms of the progression. | 83550 |
Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i < j \leq 101} x_i x_j$? | 10 |
Let $ABCD$ be a square with side length $10$. Point $P$ is located on side $BC$. By folding the paper along line $AP$, point $B$ maps to point $Q$. The line $PQ$ intersects side $CD$ at point $R$. Calculate the perimeter of triangle $PCR$. | 20 |
Three palaces, each rotating on a duck leg, make a full round in $30$, $50$, and $70$ days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards? | 525 |
Let $f(n)$ be the sum of the digits of $n$. Find $\sum_{n=1}^{99}f(n)$. | 900 |
Find the value of $|xy|$ given that $x$ and $y$ are integers and $6x^2y^2 + 5x^2 - 18y^2 = 17253$. | 54 |
In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b +... | 36 |
How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40}$ and $20^{30}$? | 2301 |
Let $N_{15}$ be the answer to problem 15. For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits. For example, if $x = 2024$, then $x' = 4022$. Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property tha... | 9 |
Find the smallest positive integer $n$ such that both the sum $n + 10$ and the product $n \times 10$ are perfect squares. | 90 |
Each of the natural numbers from $1$ to $n$ is colored either red or blue, with each color being used at least once. It turns out that: - Every red number is a sum of two distinct blue numbers; and - Every blue number is a difference between two red numbers. Determine the smallest possible value of $n$ for which such a... | 9 |
Wendy takes Honors Biology at school, a small class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday, none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched o... | 4 |
A class of $10$ students took a math test. Each problem was solved by exactly $7$ of the students. If the first nine students each solved $4$ problems, how many problems did the tenth student solve? | 6 |
Let $N$ be the greatest positive integer that can be expressed using all seven Roman numerals $I$, $V$, $X$, $L$, $C$, $D$, and $M$ exactly once each, and let $n$ be the least positive integer that can be expressed using these numerals exactly once each. Find $N - n$. Note that the arrangement $CM$ is never used in a n... | 222 |
Fred chooses a positive two-digit number with distinct nonzero digits. Laura takes Fred’s number and swaps its digits. She notices that the sum of her number and Fred’s number is a perfect square, and the positive difference between them is a perfect cube. Find the greater of the two numbers. | 74 |
Find the number of ordered pairs of integers $(m, n)$ that satisfy $20m - 10n = mn$. | 24 |
You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value, and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is $1$, the value of each blue card is equal to twice the number of red cards, and the v... | 168 |
Sam spends his days walking around the following $2 \times 2$ grid of squares: \[ \begin{array}{|c|c|} \hline 1 & 2 \\ \hline 4 & 3 \\ \hline \end{array} \] Two squares are considered adjacent if they share a side. Sam starts at the square labeled $1$ and every second, he walks to an adjacent square. How many paths can... | 167 |
Given a real number $a$, the floor of $a$, written $\lfloor a \rfloor$, is the greatest integer less than or equal to $a$. For how many real numbers $x$ such that $1 \le x \le 20$ is \[ x^2 + \lfloor 2x \rfloor = \lfloor x^2 \rfloor + 2x \? \] | 362 |
Vera has several identical matches, from which she forms a triangle. She wants any two sides of this triangle to differ in length by at least $10$ matches. However, it is impossible to create such a triangle with the available matches (without leaving any extra matches). What is the maximum number of matches Vera can h... | 62 |
Let $L$ be the number formed by 2022 digits equal to $1$, that is, $L=111\ldots111$ (2022 times). Compute the sum of the digits of the number $9L^2 + 2L$. | 4044 |
While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a \rfloor$. | 941 |
Each of the integers $1, 2, \ldots, 729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122\ldots$ How many times in this string does the substring $012$ appear? | 148 |
Consider the set $A = \{0, 1, 2, \dots , 9 \}$ and let $(B_1,B_2, \dots , B_k)$ be a collection of nonempty subsets of $A$ such that $B_i \cap B_j$ has at most two elements for $i \neq j$. What is the maximal value of $k$? | 175 |
Compute the number of natural numbers $1 \leq n \leq 10^6$ such that the least prime divisor of $n$ is $17$. | 11284 |
The side length of the largest square below is $8\sqrt{2}$, as shown. Find the area of the shaded region. \[ \text{[asy]} \] \text{size}(10\text{cm}); \text{real lsf}=0.5; \text{pen dps}=\text{linewidth}(0.7)+\text{fontsize}(10); \text{defaultpen}(dps); \text{pen ds}=\text{black}; \text{real xmin}=-18.99425911800572, \... | 64 |
Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(a,b,c,\) and \(d\) are all (not necessarily distinct) factors of \(30\) and \(abcd > 900\). | 1940 |
How many pairs of positive integers \((a, b)\) satisfy the equation \(\log_a 16 = b\)? | 3 |
The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers? | 1033 |
One can define the greatest common divisor of two positive rational numbers as follows: for $a$, $b$, $c$, and $d$ positive integers with $\gcd(a,b)=\gcd(c,d)=1$, write: \[ \gcd\left(\frac{a}{b},\frac{c}{d}\right) = \frac{\gcd(ad,bc)}{bd}. \] For all positive integers $K$, let $f(K)$ denote the number of ordered pairs ... | 1152 |
Fedja used matches to form a parallelogram, ensuring that the sides are of equal length and the vertices do not lie on a single line. It is observed that exactly 7 or 9 matches fit into the diagonals. Determine the number of matches that make up the perimeter of the parallelogram. | 22 |
Given $1994$ points on a plane, it is known that any $100$ of them can be selected such that $98$ of these can be enclosed by a circle with a diameter of $1$. Determine the minimum number of circles with a radius of $1$ required to cover all $1994$ points. | 21 |
Consider an orange and black coloring of a $20 \times 14$ square grid. Let $n$ be the number of colorings such that every row and column has an even number of orange squares. Evaluate $\log_2 n$. | 247 |
How many natural numbers exist such that their square results in a thirty-digit number with the following property: When divided from left to right into three groups of ten digits, the numbers formed by the middle group and the right group are both four times the number formed by the left group? | 18377 |
The expression $\circ 1 \circ 2 \circ 3 \circ \dots \circ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression and finds the remainder when it is divided by 2012. How many possible values are there for this remainder? | 1006 |
Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \frac{193}{137}.\] If $a+b=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$. | 28009 |
Convex pentagon $ABCDE$ satisfies $AB \parallel DE$, $BE \parallel CD$, $BC \parallel AE$, $AB = 30$, $BC = 18$, $CD = 17$, and $DE = 20$. Find its area. | 612 |
For how many integers $n$ between $1$ and $2021$ does the infinite nested expression $\sqrt{n + \sqrt{n + \sqrt{n + \sqrt{\ldots}}}}$ give a rational number? | 44 |
Consider all functions $f: \mathbb{N} \to \mathbb{N}$ satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$ and $t$ in $\mathbb{N}$. Determine the least possible value of $f(1998)$. | 120 |
Let $a$ and $b$ be positive integers such that $(2a + b)(2b + a) = 4752$. Find the value of $ab$. | 520 |
Let $P$ be the intersection point of the diagonals of a convex quadrilateral $ABCD$. It is known that the areas of triangles $\triangle ABC$, $\triangle BCD$, and $\triangle DAP$ are $8 \text{ cm}^2$, $9 \text{ cm}^2$, and $10 \text{ cm}^2$, respectively. Find the area of the quadrilateral $ABCD$. | 24 |
Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$. If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac{m}{n}$ where $m$ and $n$ are positive coprime integers, find $m+n$. | 240 |
The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\overline{BD}$, and $F$ is the midpoint of side $\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region. \[\text{[asy]} \text{size(300);} \text{defaultpen(linewidth(0.8));} \text{pair A = (0,10), B=origin, C=(12,0), D=(24,0)... | 90 |
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