prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Circle $w$ with center $O$ meets circle $\Gamma$ at points $X$ and $Y$, and the center $O$ is on circle $\Gamma$. A point $Z$ on $\Gamma$ lies outside circle $w$ such that the distances are $XZ = 11$, $OZ = 15$, and $YZ = 13$. If the radius of circle $w$ is $r$, find $r^2$. | 82 |
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute \[ \sum_{i=1}^{1000} g(i). \] | 8000 |
Determine the largest integer $n$ such that $n < 103$ and $n^3 - 1$ is divisible by $103$. | 56 |
Find the value of \[ 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 + \cdots - 2020, \] where the sign alternates between $+$ and $-$ after every three numbers. | 1010 |
A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn? | 76 |
Let $a$ and $b$ be positive integers such that $(a^3 - a^2 + 1)(b^3 - b^2 + 2) = 2020$. Find $10a + b$. | 53 |
Find a positive integer $n$ with five non-zero different digits such that $n$ is equal to the sum of all the three-digit numbers that can be formed using the digits of $n$. | 35964 |
Consider the system of equations: \[ a + 2b + 3c + \ldots + 26z = 2020 \] \[ b + 2c + 3d + \ldots + 26a = 2019 \] \[ \vdots \] \[ y + 2z + 3a + \ldots + 26x = 1996 \] \[ z + 2a + 3b + \ldots + 26y = 1995 \] where each equation is a rearrangement of the first equation with the variables cycling and the coefficients stay... | 1995 |
When the integer $({\sqrt{3} + 5})^{103} - ({\sqrt{3} - 5})^{103}$ is divided by 9, what is the remainder? | 1 |
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$. | 59 |
Determine the smallest positive integer $N$ such that there exist 6 distinct integers $a_1, a_2, a_3, a_4, a_5, a_6 > 0$ satisfying: 1. $N = a_1 + a_2 + a_3 + a_4 + a_5 + a_6$ 2. $N - a_i$ is a perfect square for $i = 1,2,3,4,5,6$. | 911 |
The infinite geometric series of positive reals $a_1, a_2, \dots$ satisfies the equation: \[ 1 = \sum_{n=1}^\infty a_n = -\frac{1}{2013} + \sum_{n=1}^{\infty} \text{GM}(a_1, a_2, \dots, a_n) = \frac{1}{N} + a_1 \] where $\text{GM}(x_1, x_2, \dots, x_k) = \sqrt[k]{x_1x_2\cdots x_k}$ denotes the geometric mean. Compute $... | 4052169 |
A boat is traveling upstream at \(5\) mph relative to the current flowing against it at \(1\) mph. If a tree branch \(10\) miles upstream from the boat falls into the current of the river, how many hours does it take to reach the boat? | 2 |
What is the smallest positive multiple of $225$ that can be written using digits $0$ and $1$ only? | 11111111100 |
Determine the largest positive integer $n$ which cannot be written as the sum of three numbers greater than $1$ that are pairwise coprime. | 17 |
Consider the sequence: $x_1 = 19$, $x_2 = 95$, $x_{n+2} = \text{lcm}(x_{n+1}, x_n) + x_n$, for $n > 1$, where $\text{lcm}(a, b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$. | 19 |
Find the rightmost non-zero digit of the expansion of $20 \times 13!$. | 6 |
A block of mass $m = \text{4.2 kg}$ slides through a frictionless table with speed $v$ and collides with a block of identical mass $m$, initially at rest, that hangs on a pendulum as shown above. The collision is perfectly elastic and the pendulum block swings up to an angle $\theta = 12^\circ$, as labeled in the diagr... | 13 |
On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$, with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets the positive difference between the numbers on the... | 4925 |
Find all 2-digit numbers $n$ having the property: The number $n^2$ is a 4-digit number of the form $\overline{xxyy}$. | 88 |
Let $x$ be the answer to this problem. For what real number $a$ is the answer to this problem also $a-x$? | 0 |
How many distinct triangles are there with prime side lengths and perimeter $100$? | 0 |
In English class, you have discovered a mysterious phenomenon: if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$, but if you spend $35$ minutes ... | 75 |
Let $f(x)$ be a function mapping real numbers to real numbers. Given that $f(f(x)) = \frac{1}{3x}$, and $f(2) = \frac{1}{9}$, find $f\left(\frac{1}{6}\right)$. | 3 |
Suppose \( xy - 5x + 2y = 30 \), where \( x \) and \( y \) are positive integers. Find the sum of all possible values of \( x \). | 31 |
Let $a$ and $b$ be positive integers satisfying $3a < b$ and $a^2 + ab + b^2 = (b + 3)^2 + 27.$ Find the minimum possible value of $a + b.$ | 25 |
In the quadrilateral $ABCD$, angles $\angle B$ and $\angle C$ are both $120^\circ$. The side lengths are given as $AB = CD = 1$ and $CB = 4$. Find the length of $AD$. | 5 |
How many ways can you mark 8 squares of an $8 \times 8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.) | 21600 |
For all positive integer triples $(m, n, k)$ such that $|m^k - n!| \le n$, find the maximum value of $\frac{n}{m}$. | 2 |
Triangle $ABC$ has side lengths $AB = 14$, $BC = 15$, and $CA = 13$. We draw a circle with diameter $AB$ such that it intersects $BC$ again at $D$ and intersects $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers, determine $100m+n... | 3308 |
Let $x, y, z$ be real numbers such that $x+y+z=-2$ and \[ (x^2+xy+y^2)(y^2+yz+z^2) + (y^2+yz+z^2)(z^2+zx+x^2) + (z^2+zx+x^2)(x^2+xy+y^2) = 625+ \frac{3}{4}(xy+yz+zx)^2. \] Compute $|xy+yz+zx|$. | 14 |
For every positive integer $n$, we define $a_{n}$ as the last digit of the sum $1 + 2 + \cdots + n$. Compute $a_{1} + a_{2} + \cdots + a_{1992}$. | 6984 |
Let $k$ be the product of every third positive integer from $2$ to $2006$, that is, \[ k = 2 \cdot 5 \cdot 8 \cdot 11 \cdots 2006. \] Find the number of zeros at the right end of the decimal representation for $k$. | 168 |
Find all positive integers $n \geq 1$ such that there exists a pair $(a, b)$ of positive integers, satisfying the following conditions: 1. $a^2 + b + 3$ is not divisible by the cube of any prime. 2. \( n = \frac{ab + 3b + 8}{a^2 + b + 3} \) | 2 |
Kara rolls a six-sided die. If on the first roll she rolls an $n$, she rolls the die $n-1$ more times. She then computes that the product of all her rolls, including the first, is $8$. How many distinct sequences of rolls could Kara have rolled? | 4 |
What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$? | 420 |
How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1, 4, 7, 10 \}$, where each number in the set is chosen at least once? For example, one possible sum is $1 + 1 + 1 + 4 + 7 + 7 + 10 + 10 = 41$. | 13 |
Let $\theta=\frac{2\pi}{2015}$, and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$, where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$. | 1441 |
A semicircle has diameter $\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\overline{AD}$, and points $E$ and $F$ lie on the arc of the semicircle. The two right triangles $\vartriangle BCF$ and $\vartriangle CDE$ are congruent. The area of $\vartriangle BCF$ is $m\sqrt{n}$, where $m$ and $n$ are positive int... | 52 |
Compute the number of five-digit positive integers $\overline{vwxyz}$ for which \[ (10v+w) + (10w+x) + (10x+y) + (10y+z) = 100. \] | 164 |
The pentagon below has three right angles. Find its area. \begin{asy} size(150); defaultpen(linewidth(1)); draw((4,10)--(0,10)--origin--(10,0)--(10,2)--cycle); label("4",(2,10),N); label("10",(0,5),W); label("10",(5,0),S); label("2",(10,1),E); label("10",(7,6),NE); \end{asy} | 76 |
Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this? | 1012 |
A set of five-digit numbers $\{N_1, ..., N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1, ..., N_k$. Find the smallest possible value of $k$. | 9 |
A positive integer $n$ is called *nice* if it has at least 3 proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is *nice* because its largest three proper divisors are $3, 2, 1$ and $6 = 3 + 2 + 1$. Find the number of *nice* integers not greater than $3000$. | 200 |
Evaluate \[ \frac{1}{729} \sum_{a=1}^{9} \sum_{b=1}^{9} \sum_{c=1}^{9} \left( abc + ab + bc + ca + a + b + c \right). \] | 215 |
Let $ABC$ be a right triangle, right at $B$, and let $M$ be the midpoint of the side $BC$. Let $P$ be the point on the bisector of the angle $\angle BAC$ such that $PM$ is perpendicular to $BC$ (with $P$ being outside the triangle $ABC$). Determine the area of triangle $ABC$ if $PM = 1$ and $MC = 5$. | 120 |
Call a pair of integers $(a,b)$ primitive if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive. | 299 |
If $a$, $b$, $c$, $d$, and $e$ are constants such that for every $x > 0$: \[ \frac{5x^4 - 8x^3 + 2x^2 + 4x + 7}{(x + 2)^4} = a + \frac{b}{x + 2} + \frac{c}{(x + 2)^2} + \frac{d}{(x + 2)^3} + \frac{e}{(x + 2)^4} \, , \] then what is the value of $a + b + c + d + e$? | 18 |
Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1, 2, \ldots, ab$, putting the numbers $1, 2, \ldots, b$ in the first row, $b + 1, b + 2, \ldots, 2b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting ... | 21 |
Given positive integers $a$, $b$, and $c$ such that $a + b + c = 20$, determine the number of possible integer values for $\frac{a + b}{c}$. | 5 |
Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$. If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$? | 1002 |
Given that the base-$17$ integer $\overline{8323a02421_{17}}$ (where $a$ is a base-$17$ digit) is divisible by $\overline{16_{10}}$, find $a$. Express your answer in base $10$. | 7 |
GM Bisain's IQ is so high that he can move around in $10$-dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates. | 88660 |
Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$, the number $d$ is a divisor of $\overline{aabbcc}$ if and only if $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. Note: The numbers $a \ne 0$, $b$, and $c$ need not be differe... | 99 |
Find all positive integers $n$ that have 4 digits, all of which are perfect squares, and such that $n$ is divisible by 2, 3, 5, and 7. | 4410 |
Define the \textit{bigness} of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and \textit{bigness} $N$ and another one with integer side lengths and \textit{bigness} $... | 55 |
Given that there are $24$ primes between $3$ and $100$, inclusive, determine the number of ordered pairs $(p, a)$ where $p$ is a prime number such that $3 \le p < 100$, and $1 \le a < p$. The condition is that the sum: \[ a + a^2 + a^3 + \cdots + a^{(p-2)!} \] is not divisible by $p$. | 27 |
Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain? | 94 |
There are 16 cities participating in a football match. Each city has two teams: Team A and Team B. According to the rules, each team can compete with any other team from different cities at most once. After a few days, it is observed that the number of matches played by each team, except for Team A from City 1, is uniq... | 15 |
If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by \[ S(n) = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor + 10 \left( n - 10^{\lfloor \log n \rfloor} \cdot \left\lfloor \frac{n}{10^{\lfloo... | 108 |
Find the number of subsets of $\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$ where the sum of the elements in the subset is $49$. | 22 |
Pairwise distinct points $P_1,P_2,\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\lvert P_iP_{i+1} \rvert = 1$ for $i=1,2,\ldots, 16$. (We take $P_{17}$ to be the point $P_1$.) We construct points $Q_1,Q_2,\ldots,Q_{16}$ as follows: for each $i$, a fair coin is flipped.... | 88 |
Find the smallest positive integer $n$ such that $\underbrace{2^{2^{\cdots^{2}}}}_{n} > 3^{3^{3^3}}$. (The notation $\underbrace{2^{2^{\cdots^{2}}}}_{n}$ is used to denote a power tower with $n$ $2$’s. For example, $\underbrace{2^{2^{\cdots^{2}}}}_{4}$ would equal $2^{2^{2^2}}$.) | 5 |
Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$? | 360 |
Let $a$, $b$, $c$ be positive integers such that $29a + 30b + 31c = 366$. Find $19a + 20b + 21c$. | 246 |
Find the 1000th number, in ascending order, that is coprime to $105$. | 2186 |
For every three-digit natural number $n$ (where the leading digit of $n$ is nonzero), consider the number $n_0$ obtained by eliminating all zero digits from $n$. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$ for which $n_0$ is a divisor of $n$ and $n_0 \neq n$. | 93 |
Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What i... | 26 |
Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$, determine the largest possible value of $y$. | 4 |
Compute the number of ordered triples \((x, y, z)\) of integers satisfying the equation: \[ x^2 + y^2 + z^2 = 9. \] | 30 |
Suppose the polynomial $f(x) = x^{2014}$ is expressed as $f(x) = \sum^{2014}_{k=0} a_k \binom{x}{k}$ for some real numbers $a_0, a_1, \ldots, a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$. | 2004 |
Let $s_1, s_2, s_3$ be the three roots of the polynomial $x^3 + x^2 + \frac{9}{2}x + 9$. Find the value of $a + b + c$ if the expression $\prod_{i=1}^{3}(4s^4_i + 81)$ can be written as $2^a3^b5^c$. | 16 |
For a positive integer $n$, an $n$-branch $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to appear in $B$ if it is an element of the last set $S_m$. Define an $... | 76 |
Given that $20^{22} + 1$ has exactly four prime divisors $p_1 < p_2 < p_3 < p_4$, determine $p_1 + p_2$. | 490 |
Two ants crawl along the sides of the $49$ squares of a $7 \times 7$ board. Each ant passes through all $64$ vertices exactly once and returns to its starting point. What is the smallest possible number of sides covered by both ants? | 16 |
Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x) = x^{3} + px^{2} + qx + r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime p... | 551 |
Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$. We want to compute, using a computer, the extr... | 75 |
The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 231361 |
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? \[ \textbf{(A)}\ 37 \quad \textbf{(B)}\ 63 \quad \textbf{(C)}\ 117 \quad \textbf{(D)}\ 143 \quad \textbf{... | 143 |
Find the largest natural number $n$ for which there exist different sets $S_1, S_2, \ldots, S_n$ such that: 1. $|S_i \cup S_j| \leq 2004$ for each pair $1 \leq i, j \leq n$. 2. $S_i \cup S_j \cup S_k = \{1, 2, \ldots, 2008\}$ for each three integers $1 \leq i < j < k \leq n$. | 32 |
A marble is placed on each $33$ unit square of a $10 \times 10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board? | 438 |
Let the roots of \(x^{2022} - 7x^{2021} + 8x^2 + 4x + 2\) be \(r_1, r_2, \ldots, r_{2022}\), the roots of \(x^{2022} - 8x^{2021} + 27x^2 + 9x + 3\) be \(s_1, s_2, \ldots, s_{2022}\), and the roots of \(x^{2022} - 9x^{2021} + 64x^2 + 16x + 4\) be \(t_1, t_2, \ldots, t_{2022}\). Compute the value of: \[ \sum_{1 \le i,j \... | 191 |
For all positive integers $n$, let \[ f(n) = \sum_{k=1}^n \varphi(k) \left\lfloor \frac{n}{k} \right\rfloor^2. \] Compute $f(2019) - f(2018)$. Here, $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$. | 11431 |
For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2 y}$. The sum of the solutions to the equation \[ 4096 f(f(x, x), x) = x^{13} \] can be written in simplest form as $\frac{m}{n}$. Compute $m + n$. | 177 |
A subset $X$ of $\{1, 2, 3, \ldots, 10000\}$ has the following property: If $a, b$ are distinct elements of $X$, then $ab \not\in X$. What is the maximal number of elements in $X$? | 9901 |
Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight \(2015\) pounds? | 13 |
Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[ \left\lfloor\frac{2008^1}{1}\right\rfloor, \left\lfloor\frac{2008^2}{2}\right\rfloor, \left\lfloor\frac{2008^3}{3}\right\rfloor, \left\lfloor\frac{2008^4}{4}\right\rfloor, \ldots. \] | 2003 |
Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$. | 20171928 |
Compute the remainder when $2^{30}$ is divided by $1000$. | 824 |
Let $\alpha$ and $\beta$ be positive integers such that $\frac{43}{197} < \frac{\alpha}{\beta} < \frac{17}{77}$. Find the minimum possible value of $\beta$. | 32 |
A two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers. | 5 |
If $57a + 88b + 125c \geq 1148$, where $a, b, c > 0$, what is the minimum value of \[ a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2? \] | 466 |
Let $f(n)$ denote the largest odd factor of $n$, including possibly $n$ itself. Determine the value of \[ \frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048}, \] rounded to the nearest integer. | 1365 |
Consider the following three lines in the Cartesian plane: \[ \begin{cases} \ell_1: & 2x - y = 7\\ \ell_2: & 5x + y = 42\\ \ell_3: & x + y = 14 \end{cases} \] Let \( f_i(P) \) correspond to the reflection of the point \( P \) across \( \ell_i \). Suppose \( X \) and \( Y \) are points on the \( x \) and \( y \) axes, r... | 260 |
Find the maximal positive integer $p$ such that $5^7$ is the sum of $p$ consecutive positive integers. | 250 |
Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$. | 665 |
Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients \textit{equivalent} if $p(\alpha)\equiv q(\alpha)\pmod{3}$. It is known that every such polynomial is equivalent to exactly one of $0, 1, x, x^2, \ldots, x^{727}$. Find the largest integer $n < 728$ for which there exist... | 727 |
How many ways are there to fill in a $2 \times 2$ square grid with the numbers $1, 2, 3,$ and $4$ such that the numbers in any two grid squares that share an edge have an absolute difference of at most $2$? | 8 |
Find the $2016$th smallest positive integer that satisfies the congruence $x^x \equiv x \pmod{5}$. | 3360 |
How many positive integers $n$ are there such that $n+2$ divides $(n+18)^2$? | 7 |
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