prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Let $x$ be a real number such that: \[ 4^{2x} + 2^{-x} + 1 = (129 + 8\sqrt{2})(4^{x} + 2^{-x} - 2^{x}). \] Find $10x$. | 35 |
Define a sequence of integers $a_1, a_2, \dots, a_k$ where every term $a_i \in \{1,2\}$, and let $S$ denote their sum. Another sequence of integers $t_1, t_2, \ldots, t_k$ is defined by \[t_i=\sqrt{a_i(S-a_i)},\] for all $t_i$. Suppose that \(\sum_{1 \leq i \leq k} t_i=4000.\) Find the value of \(\sum_{1 \leq i \leq k}... | 476 |
In triangle $ABC$, angle $A$ is twice angle $B$, angle $C$ is obtuse, and the three side lengths $a, b, c$ are integers. Determine, with proof, the minimum possible perimeter. | 77 |
The diagram shows a $20 \times 20$ square $ABCD$. The points $E$, $F$, and $G$ are equally spaced on side $BC$. The points $H$, $I$, $J$, and $K$ on side $DA$ are placed so that the triangles $BKE$, $EJF$, $FIG$, and $GHC$ are isosceles. Points $L$ and $M$ are midpoints of the sides $AB$ and $CD$, respectively. Find th... | 200 |
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $\frac{n(n+1)}{2}$. A Pythagorean triple of \textit{square numbers} is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of \textit{triangular numbers} (a PTTN) be an ordered triple of positive integers $... | 14 |
A week ago, Sandy's seasonal Little League batting average was $360$. After five more at bats this week, Sandy's batting average increased to $400$. What is the smallest number of hits that Sandy could have had this season? | 12 |
Let $x$ be a positive real number. Define \[ A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad \text{and} \quad C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}. \] Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$. | 183 |
Let \( G, O, D, I, \) and \( T \) be digits that satisfy the following equation: \[ \begin{array}{cccc} & G & O & G & O \\ + & D & I & D & I \\ \hline G & O & D & O & T \end{array} \] (Note that \( G \) and \( D \) cannot be \( 0 \), and that the five variables are not necessarily different.) Compute the value of GODOT... | 10908 |
A classroom has $30$ students, each of whom is either male or female. For every student $S$, we define their ratio to be the number of students of the opposite gender as $S$ divided by the number of students of the same gender as $S$ (including $S$). Let $\Sigma$ denote the sum of the ratios of all $30$ students. Find ... | 2 |
There are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair ... | 50 |
In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has an area of $36\pi$, find the perimeter of the triangle. | 36 |
Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set \{0,1,\dots,2010\}. Call such a polynomial "good" if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20$, $P(m_... | 460 |
Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks an integer $c \in \{2,3,\dots,p-1\}$. Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper. Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f... | 65819 |
A function $f$ has its domain equal to the set of integers $0, 1, \ldots, 11$, and $f(n) \geq 0$ for all such $n$. The function $f$ satisfies the following conditions: - $f(0) = 0$ - $f(6) = 1$ - If $x \geq 0$, $y \geq 0$, and $x + y \leq 11$, then $f(x + y) = \frac{f(x) + f(y)}{1 - f(x)f(y)}$ Find $f(2)^2 + f(10)^2$. | 14 |
Let $T$ denote the 15-element set $\{10a+b : a,b \in \mathbb{Z}, 1 \le a < b \le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$. | 9 |
Let $f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a z_b + z_c z_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$. | 8 |
Suppose $2015 = a_1 < a_2 < a_3 < \cdots < a_k$ is a finite sequence of positive integers, and for all $m, n \in \mathbb{N}$ and $1 \le m, n \le k$, the inequality $$a_m + a_n \ge a_{m+n} + |m-n|$$ holds. Determine the largest possible value $k$ can obtain. | 2016 |
Let $\{a_n\}$ be a sequence of integers satisfying the recurrence relation: \[ (n-1)a_{n+1} = (n+1)a_n - 2(n-1) \quad \forall n \ge 1. \] Given that $2000 \mid a_{1999}$, determine the smallest integer $n \ge 2$ such that $2000 \mid a_n$. | 249 |
Find the smallest constant $c$ such that for every 4 points in a unit square, there are two points with a distance less than or equal to $c$ apart. | 1 |
For how many integers $1 \leq k \leq 2013$ does the decimal representation of $k^k$ end with a $1$? | 202 |
Sherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability $\frac{1}{2}$. When she reaches 10, she stops. What is the expected number (average number) of steps that Sherry will take... | 81 |
An $80 \times 80$ grid is colored orange and black. A square is black if and only if either the square below it or the square to the left of it is black, but not both. If there is no such square, consider it as if it were orange. The only exception is the bottom left square, which is black. Consider the diagonal from t... | 40 |
The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$, and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $\triangle ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16$, $r_B = 25$, and $r_C = 36$, determine the radius of... | 74 |
The faces of a cube contain the numbers 1, 2, 3, 4, 5, 6 such that the sum of the numbers on each pair of opposite faces is 7. For each of the cube’s eight corners, multiply the three numbers on the faces incident to that corner, and write down its value. What is the sum of the eight values assigned to the cube’s corne... | 343 |
Find the smallest positive integer $N$ such that $N$ is divisible by $19$, and when the digits of $N$ are read in reverse order, the result (after removing any leading zeroes) is divisible by $36$. | 2394 |
Let $F_k(a,b) = (a+b)^k - a^k - b^k$ and let $S = \{1,2,3,4,5,6,7,8,9,10\}$. For how many ordered pairs $(a,b)$ with $a,b \in S$ and $a \leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer? | 22 |
We call $A_1, A_2, \ldots, A_n$ an $n$-division of $A$ if: 1. $A_1 \cap A_2 \cap \cdots \cap A_n = A$, 2. $A_i \cap A_j \neq \emptyset$ for $i \neq j$. Find the smallest positive integer $m$ such that for any $14$-division $A_1, A_2, \ldots, A_{14}$ of $A = \{1, 2, \ldots, m\}$, there exists a set $A_i$ ($1 \leq i \leq... | 56 |
Let $f(n)$ be the integer closest to $\sqrt{n}$. Compute the largest $N$ less than or equal to $2018$ such that $\sum_{i=1}^N\frac{1}{f(i)}$ is integral. | 1980 |
Given a $2 \times 2$ tile and seven dominoes ($2 \times 1$ tile), find the number of ways to tile a $2 \times 7$ rectangle using some of these tiles without leaving gaps or overlapping any two tiles. | 59 |
There are infinitely many boxes - initially, one of them contains $n$ balls, and all others are empty. In a single move, we take some balls from a non-empty box and put them into an empty box. On a sheet of paper, we write down the product of the resulting number of balls in the two boxes. After some moves, the sum of ... | 65 |
Let $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$, $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$, $C(x)=x^3-6x^2+5x+15$, $H(x)=x^4+2x^3+3x^2+4x+5$, $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$, $N(x)=\text{the number of integers that divide }\left\lfloor x\right\rfloor$, ... | 3 |
Consider all ordered pairs \((m, n)\) of positive integers satisfying the equation \(59m - 68n = mn\). Find the sum of all possible values of \(n\) in these ordered pairs. | 237 |
The familiar $3$-dimensional cube has $6$ $2$-dimensional faces, $12$ $1$-dimensional edges, and $8$ $0$-dimensional vertices. Find the number of $9$-dimensional sub-subfaces in a $12$-dimensional cube. | 1760 |
Let $\triangle ABC$ satisfy $AB = 17$, $AC = \frac{70}{3}$, and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with... | 22 |
Let $S$ be a set of $2020$ distinct points in the plane. Define \[M=\{P : P \text{ is the midpoint of } XY \text{ for some distinct points } X, Y \text{ in } S\}.\] Find the least possible value of the number of points in $M$. | 4037 |
Determine the maximum possible value of \[ \frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4} \] over all non-zero real numbers $x$. | 576 |
You are playing a game called "Hovse." Initially, you have the number $0$ on a blackboard. At any moment, if the number $x$ is written on the board, you can perform one of the following operations: - Replace $x$ with $3x + 1$ - Replace $x$ with $9x + 1$ - Replace $x$ with $27x + 3$ - Replace $x$ with $\left \lfloor \fr... | 127 |
Let $a$ and $b$ be positive real numbers such that $(65a^2 + 2ab + b^2)(a^2 + 8ab + 65b^2) = (8a^2 + 39ab + 7b^2)^2$. Then one possible value of $\frac{a}{b}$ satisfies $2 \left(\frac{a}{b}\right) = m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$. | 2636 |
Let $T$ be a triangle whose vertices have integer coordinates, such that each side of $T$ contains exactly $m$ points with integer coordinates. If the area of $T$ is less than $2020$, determine the largest possible value of $m$. | 64 |
Let $n$ be a square number with 4 digits, such that all its digits are less than 6. If we add 1 to each digit, the resulting number is another square. Find $n$. | 2025 |
Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$. A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA_1$, $BB_1$, and $DD_1$ of the prism. Calculate $r$. | 4 |
It is known that $a^{2005} + b^{2005}$ can be expressed as a polynomial of $a + b$ and $ab$. Find the sum of the coefficients of this polynomial. | 1 |
Let $ABC$ be an isosceles triangle with $\angle A = 90^\circ$. Points $D$ and $E$ are selected on sides $AB$ and $AC$, and points $X$ and $Y$ are the feet of the altitudes from $D$ and $E$ to side $BC$. Given that $AD = 48\sqrt{2}$ and $AE = 52\sqrt{2}$, compute $XY$. | 100 |
Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference. | 247 |
A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off the square's perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 2021 regions. Compute the smallest possible value of $n$. | 63 |
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of the number of divisors of $2p^2+2p+1$. | 2 |
Consider a triangle $ABC$ with $AB = 13$, $BC = 14$, $CA = 15$. A line perpendicular to $BC$ divides the interior of $\triangle ABC$ into two regions of equal area. Suppose that the aforesaid perpendicular cuts $BC$ at $D$, and cuts $\triangle ABC$ again at $E$. If $L$ is the length of the line segment $DE$, find $L^2$... | 112 |
$ABCDEF GH$ is a regular octagon with a side length of $10$ units. Consider a circle with center $A$ and radius $AC$, which intersects another circle with center $D$ and radius $CD$ at a point $I$, different from $C$. What is the length of the segment $IF$? | 10 |
The roots of the polynomial $f(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1$ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has a positive imaginary part. Let $S$ be the sum of the values of nice ... | 31 |
The sweeties shop called "Olympiad" sells boxes of $6$, $9$, or $20$ chocolates. Groups of students from a nearby school collect money to buy a chocolate for each student; they do this by buying a box and then distributing a chocolate to everyone. For example, students can form groups of $15 = 6 + 9$ students or $38 = ... | 43 |
Find the number of ordered triples of positive integers \((a, b, c)\) such that \(abc\) divides \((ab + 1)(bc + 1)(ca + 1)\). | 16 |
On the board, we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold: 1. The sum of every $40$ consecutive numbers is equal to $0$. 2. The sum of every $42$ consecutive numbers is not equal to $0$. We denote by $S_n$ the sum... | 20 |
Find the sum of all $x$ from $2$ to $1000$ inclusive such that \( \prod_{n=2}^x \log_{n^n}(n+1)^{n+2} \) is an integer. | 739 |
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right]$. Here $[x]$ denotes the greatest integer less than or equal to $x$. | 2997 |
A bitstring of length $\ell$ is a sequence of $\ell$ $0$'s or $1$'s in a row. How many bitstrings of length $2014$ have at least $2012$ consecutive $0$'s or $1$'s? | 16 |
Let $a, b, c$ be three distinct positive integers such that the sum of any two of them is a perfect square. Find the minimal sum $a + b + c$. | 55 |
If $x$ is a real number such that $3^x = 27x$, compute $\log_3 \left(\frac{3^{3^x}}{x^{3^3}}\right)$. | 81 |
Initially, there are $3^{80}$ particles at the origin $(0,0)$. At each step, the particles are moved to points above the $x$-axis as follows: if there are $n$ particles at any point $(x,y)$, then $\left\lfloor \frac{n}{3} \right\rfloor$ of them are moved to $(x+1,y+1)$, $\left\lfloor \frac{n}{3} \right\rfloor$ are move... | 80 |
A cross is the figure composed of $6$ unit squares as shown below (and any figure made of it by rotation).  Find the greatest number of crosses that can be cut from a $6 \times 11$ sheet of paper divided into u... | 8 |
Suppose that $a$ and $b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $q$. Determine the minimum value of $p + q$. | 296 |
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Let $D$, $E$, and $F$ be the feet of the altitude, angle bisector, and median from $A$ to $BC$, respectively. If $DE = 3$ and $EF = 5$, compute the length of $BC$. | 20 |
Derek fills a square $10 \times 10$ grid with $50$ $1$s and $50$ $2$s. He takes the product of the numbers in each of the $10$ rows. He takes the product of the numbers in each of the $10$ columns. He then sums these $20$ products up to get an integer $N$. Find the minimum possible value of $N$. | 640 |
Find the smallest positive integer $n$ for which the polynomial \[x^n-x^{n-1}-x^{n-2}-\cdots -x-1\] has a real root greater than $1.999$. | 10 |
Let $f(x)=\frac{9^x}{9^x+3}$. Compute $\sum_{k} f \left( \frac{k}{2002} \right)$, where $k$ goes over all integers $k$ between $0$ and $2002$ which are coprime to $2002$. | 360 |
Let $a$ be a positive real number and $n$ a non-negative integer. Determine $S-T$, where $$ S = \sum_{k=-2n}^{2n+1} \frac{(k-1)^2}{a^{| \lfloor \frac{k}{2} \rfloor |}} $$ and $$ T = \sum_{k=-2n}^{2n+1} \frac{k^2}{a^{| \lfloor \frac{k}{2} \rfloor |}} $$ | 0 |
Two integers are called *relatively prime* if they share no common factors other than $1$. Determine the sum of all positive integers less than $162$ that are relatively prime to $162$. | 4374 |
Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is $0.75$. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is $0.25$. He decides not to make it rain today. Find the smallest ... | 9 |
Let $A=(a_{ij})$ be the $n \times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j + j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A \neq 0$. | 5 |
To each element of the set $S = \{1, 2, \ldots, 1000\}$ a color is assigned. Suppose that for any two elements $a, b$ of $S$, if $15$ divides $a + b$, then they are both assigned the same color. What is the maximum possible number of distinct colors used? | 8 |
Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \neq 0$. | 19 |
Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is friends with four, Elías is friends with five, and so on, with each subsequent person being friends with one more person than the previous person. This pattern continues ... | 13 |
Find the maximum positive integer $k$ such that for any positive integers $m,n$ where $m^3+n^3>(m+n)^2$, we have: \[ m^3+n^3\geq (m+n)^2+k \] | 10 |
Al and Bill play a game involving a fair six-sided die. The die is rolled until either a number less than $5$ is rolled on consecutive tosses, or a number greater than $4$ is rolled on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be relative... | 29 |
Henry rolls a fair die. If the die shows the number $k$, Henry will then roll the die $k$ more times. The probability that Henry will never roll a 3 or a 6 either on his first roll or on one of the $k$ subsequent rolls is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 904 |
Given a board with size $25 \times 25$. Some $1 \times 1$ squares are marked, so that for each $13 \times 13$ and $4 \times 4$ sub-boards, there are at least $\frac{1}{2}$ of the squares in the sub-board that are marked. Find the least possible number of marked squares in the entire board. | 313 |
Jacob's analog clock has 12 equally spaced tick marks on the perimeter, but all the digits have been erased, so he doesn't know which tick mark corresponds to which hour. Jacob takes an arbitrary tick mark and measures clockwise to the hour hand and minute hand. He measures that the minute hand is 300 degrees clockwise... | 500 |
Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \le n \le 50$ such that $n$ divides $\phi^{!}(n)+1$. | 30 |
9. What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length $12$? | 84 |
Let $S = \{1, 2, 3, \ldots, 280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime. | 217 |
Find the integer closest to \[ \frac{1}{\sqrt[4]{5^4+1} - \sqrt[4]{5^4-1}} \] | 250 |
Consider the following one-person game: A player starts with a score $0$ and writes the number $20$ on an empty whiteboard. At each step, she may erase any one integer (call it $a$) and write two positive integers (call them $b$ and $c$) such that $b + c = a$. The player then adds $b \times c$ to her score. She repeats... | 0 |
Define $f(n) = \frac{n^2 + n}{2}$. Compute the number of positive integers $n$ such that $f(n) \leq 1000$ and $f(n)$ is the product of two prime numbers. | 8 |
Find the sum of the positive integer solutions to the equation $\left\lfloor\sqrt[3]{x}\right\rfloor + \left\lfloor\sqrt[4]{x}\right\rfloor = 4.$ | 231 |
You roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of the sum of the three dice can be written as $\frac{a}{b}$ in simplest form. Find $a+b$. | 706 |
Let $AC$ be a diameter of a circle $\omega$ with radius $1$. Let $D$ be a point on $AC$ such that $CD = \frac{1}{5}$. Let $B$ be the point on $\omega$ such that $DB$ is perpendicular to $AC$, and let $E$ be the midpoint of $DB$. The line tangent to $\omega$ at $B$ intersects line $CE$ at point $X$. Compute $AX$. | 3 |
$A$ and $B$ play a game where $A$ chooses a positive integer $n \in \{1, 2, \dots, 1001\} = S$. $B$ must guess the value of $n$ by choosing several subsets of $S$. $A$ will then tell $B$ how many of these subsets contain $n$. $B$ will perform this action three times, selecting $k_1$, $k_2$, and $k_3$ subsets of $S$ res... | 28 |
Let $ABCD$ be a convex cyclic quadrilateral. Suppose $P$ is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of $ABCD$ is the least. If $\{PA, PB, PC, PD\} = \{3, 4, 6, 8\}$, what is the maximum possible area of $ABCD$? | 54 |
Given in the figure: - The line through $E$ and $F$ is parallel to the line passing through $A$ and $B$. - The area of the triangle $\triangle FEC$ is $40$ cm$^2$. - The sum of the areas of the triangles $\triangle ADF$ and $\triangle DBE$ is $150$ cm$^2$. Determine whether the area of the triangle $\triangle DEF$ foll... | 60 |
The incircle of $\triangle ABC$ is tangent to sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$. The sides of $\triangle ABC$ have lengths $AB = 73$, $BC = 123$, and $AC = 120$. Find the length $EG$. | 119 |
Find the least integer $n$ greater than $345$ such that $\frac{3n+4}{5}$, $\frac{4n+5}{3}$, and $\frac{5n+3}{4}$ are all integers. | 397 |
In a triangle $ABC$, let $I$ denote the incenter. The lines $AI$, $BI$, and $CI$ intersect the incircle at points $P$, $Q$, and $R$, respectively. Given that $\angle BAC = 40^\circ$, determine the value of $\angle QPR$ in degrees. | 55 |
Compute the following: $$\sum^{99}_{x=0} (x^2 + 1)^{-1} \pmod{199}$$ where $x^{-1}$ is the value $0 \le y \le 199$ such that $xy - 1$ is divisible by $199$. | 100 |
We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2 \equiv a \pmod{n}$, $x^3 \equiv a \pmod{n}$, or $x^5 \equiv a \pmod{n}$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of resi... | 694 |
A frog is hopping from $(0,0)$ to $(8,8)$. The frog can hop from $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. The frog is only allowed to hop to point $(x,y)$ if $|y-x| \leq 1$. Compute the number of distinct valid paths the frog can take. | 256 |
A pentagon has four interior angles each equal to $110^\circ$. Find the degree measure of the fifth interior angle. | 100 |
Evaluate the following expression: \[2013^2 + 2011^2 + \ldots + 5^2 + 3^2 - 2012^2 - 2010^2 - \ldots - 4^2 - 2^2\] | 2027090 |
How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$? | 399 |
Find the largest integer $n$ for which $2^n$ divides \[ \binom{2}{1} \binom{4}{2} \binom{6}{3} \dots \binom{128}{64}. \] | 193 |
Given a triangle $ABC$, consider the semicircle with diameter $\overline{EF}$ on $\overline{BC}$ tangent to $\overline{AB}$ and $\overline{AC}$. If $BE=1$, $EF=24$, and $FC=3$, find the perimeter of $\triangle{ABC}$. | 84 |
There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points that are exactly $1$ unit apart. Find the maximum possible value of $I$. | 3954 |
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