We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating.

Find the greatest seven-digit integer divisible by $132$ whose digits, in order, are $2, 0, x, y, 1, 2, z$ where $x$, $y$, and $z$ are single digits.

In a school class with $ 3n$ children, any two children make a common present to exactly one other child. Prove that for all odd $ n$ it is possible that the following holds: For any three children $ A$, $ B$ and $ C$ in the class, if $ A$ and $ B$ make a present to $ C$ then $ A$ and $ C$ make a present to $ B$.

Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.

For each prime number $p$, determine the number of residue classes modulo $p$ which can be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers. [i](Daniel Holmes)[/i]

Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.

Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

For positive integers $n,$ let $\nu_3 (n)$ denote the largest integer $k$ such that $3^k$ divides $n.$ Find the number of subsets $S$ (possibly containing 0 or 1 elements) of $\{1, 2, \ldots, 81\}$ such that for any distinct $a,b \in S$, $\nu_3 (a-b)$ is even. [i]Author: Alex Zhu[/i] [hide="Clarification"]We only need $\nu_3(a-b)$ to be even for $a>b$. [/hide]

Prove that for positive integers $ x,y,z$ the number $ x^2 \plus{} y^2 \plus{} z^2$ is not divisible by $ 3(xy \plus{} yz \plus{} zx)$.

Find the units digit in the decimal expansion of \[\left(2008+\sqrt{4032000}\right)^{2000}+\left(2008+\sqrt{4032000}\right)^{2001}+\left(2008+\sqrt{4032000}\right)^{2002}+\]\[\cdots+\left(2008+\sqrt{4032000}\right)^{2007}+\left(2008+\sqrt{4032000}\right)^{2008}.\]

Find all integer solutions to $2 x^4 + 1 = y^2.$

Which of the following is the set of all perfect squares that can be written as sum of three odd composite numbers? $\textbf{a)}\ \{(2k + 1)^2 : k \geq  0\}$ $\textbf{b)}\ \{(4k + 3)^2 : k \geq  1\}$ $\textbf{c)}\ \{(2k + 1)^2 : k \geq  3\}$ $\textbf{d)}\ \{(4k + 1)^2 : k \geq 2\}$ $\textbf{e)}\ \text{None of above}$

Solve the equation $x^2 +7=2^n$ in integers.

How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.

Prove that for each positive integer $ n$, there are pairwise relatively prime integers $ k_0,k_1,\ldots,k_n$, all strictly greater than $ 1$, such that $ k_0k_1\ldots k_n\minus{}1$ is the product of two consecutive integers.

Let $n$ be a positive integer. Find the greatest common divisor of the numbers $\binom{2n}{1},\binom{2n}{3},\binom{2n}{5},...,\binom{2n}{2n-1}$.

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $ 4$, and the second number is divided by $ 5$. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square? [asy]unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(15pt)); draw(Circle(origin,1)); for(int i = 0;i < 6; ++i) { draw(origin--dir(60i+30)); } label("$7$",midpoint(origin--(dir(0))),E); label("$1$",midpoint(origin--(dir(60))),NE); label("$6$",midpoint(origin--(dir(120))),NW); label("$3$",midpoint(origin--(dir(180))),W); label("$9$",midpoint(origin--(dir(240))),SW); label("$2$",midpoint(origin--(dir(300))),SE); draw((2,0)--(3.5,0)--(3.5,1)--(2,1)--cycle); draw((2,0)--(3.5,0)--(3.5,-1)--(2,-1)--cycle); pair[] V = {(2.5,0.5),(2,0),(3,0),(2.5,-0.5),(2,-1),(3,-1)}; for(int i = 0; i <= 5; ++i) { pair A = V[i]; path p = A--(A.x,A.y + 0.5)--(A.x + 0.5,A.y + 0.5)--(A.x + 0.5, A.y)--cycle; fill(p,mediumgray); draw(p); } path pointer = (-2.5,-0.125)--(-2.5,0.125)--(-1.2,0.125)--(-1.05,0)--(-1.2,-0.125)--cycle; fill(pointer,mediumgray); draw(pointer); label("Pointer",(-1.85,0),fontsize(10pt)); label("$4$",(2,0.5),2N + 2W); label("$3$",(2,0),2N + 2W); label("$2$",(2,-0.5),2N + 2W); label("$1$",(2,-1),2N + 2W); label("$1$",(2,-1),2S + 2E); label("$2$",(2.5,-1),2S + 2E); label("$3$",(3,-1),2S + 2E);[/asy]$ \textbf{(A)}\ \frac {1}{3}\qquad \textbf{(B)}\ \frac {4}{9}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {5}{9}\qquad \textbf{(E)}\ \frac {2}{3}$

Let $\{a_n\}$ be a sequence of integers satisfying the following conditions. [list] [*] $a_1=2021^{2021}$ [*] $0 \le a_k < k$ for all integers $k \ge 2$ [*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$. [/list] Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.

Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\] [i]Timothy Chu.[/i]

What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?

$p$ is an odd prime number. Prove that there exists a natural number $x$ such that $x$ and $4x$ are both primitive roots modulo $p$. [i]Proposed by Mohammad Gharakhani[/i]

For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$. [i]Romania[/i]

How many primes $p$ are there such that the number of positive divisors of $p^2+23$ is equal to $14$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of above} $

For each $ k\in\mathbb{N} ,k\le 2000, $ Let $ r_k $ be the remainder of the division of $ k $ by $ 4, $ and $ r'_k $ be the remainder of the division of $ k $ by $ 3. $ Prove that there is an unique $ m\in\mathbb{N} ,m\le 1999 $ such that $$ r_1+r_2+\cdots +r_m=r'_{m+1} +r'_{m+2} +\cdots r'_{2000} . $$ [i]Mircea Fianu[/i]