2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations: [list] [*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip. [*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list] Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it. [i]Brian Hamrick.[/i]

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

Let $p$ be an odd prime. Prove that: \[\displaystyle\sum_{k\equal{}1}^{p\minus{}1}k^{2p\minus{}1} \equiv \frac{p(p\plus{}1)}{2} \pmod{p^2}\]

Let $p$ be a prime number that is greater than $3$. Show that there exist some integers $a_{1}, a_{2}, \cdots a_{k}$ that satisfy: \[-\frac{p}{2}< a_{1}< a_{2}< \cdots <a_{k}< \frac{p}{2}\] making the product: \[\frac{p-a_{1}}{|a_{1}|}\cdot \frac{p-a_{2}}{|a_{2}|}\cdots \frac{p-a_{k}}{|a_{k}|}\] equals to $3^{m}$ where $m$ is a positive integer.

Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $

For a given rational number $r$, find all integers $z$ such that \[2^z + 2 = r^2\mbox{.}\] [i](Swiss Mathematical Olympiad 2011, Final round, problem 7)[/i]

Let $ a_1,a_2,\dots$ be a sequence of integers determined by the rule $ a_n\equal{}a_{n\minus{}1}/2$ if $ a_{n\minus{}1}$ is even and $ a_n\equal{}3a_{n\minus{}1}\plus{}1$ if $ a_{n\minus{}1}$ is odd. For how many positive integers $ a_1 \le 2008$ is it true that $ a_1$ is less than each of $ a_2$, $ a_3$, and $ a_4$? $ \textbf{(A)}\ 250 \qquad \textbf{(B)}\ 251 \qquad \textbf{(C)}\ 501 \qquad \textbf{(D)}\ 502 \qquad \textbf{(E)}\ 1004$

Prove that there exists a succession $a_1, a_2, ... , a_k, ...$, where each $a_i$ is a digit ($a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)$ ) and $a_0=6$, such that, for each positive integrer $n$, the number $x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1}$ verify that $x_n^2-x_n$ is divisible by $10^n$.

A sequence $a_1,a_2,a_3,\ldots $ of non-negative integers is such that $a_{n+1}$ is the last digit of $a_n^n+a_{n-1}$ for all $n>2$. Is it always true that for some $n_0$ the sequence $a_{n_0},a_{n_0+1},a_{n_0+2},\ldots$ is periodic?

We define $N$ as the set of natural numbers $n<10^6$ with the following property: There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$. Find $|N|$.

Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.

Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively. a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint? b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint? Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and for $X,Y \subseteq \mathbb{Q}$.

Let $a,b$ be coprime positive integers. Show that the number of positive integers $n$ for which the equation $ax+by=n$ has no positive integer solutions is equal to $\frac{(a-1)(b-1)}{2}-1$.

At ARML, Santa is asked to give rubber duckies to $2013$ students, one for each student. The students are conveniently numbered $1,2,\cdots,2013$, and for any integers $1 \le m < n \le 2013$, students $m$ and $n$ are friends if and only if $0 \le n-2m \le 1$. Santa has only four different colors of duckies, but because he wants each student to feel special, he decides to give duckies of different colors to any two students who are either friends or who share a common friend. Let $N$ denote the number of ways in which he can select a color for each student. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Lewis Chen[/i]

Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$.

Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) $f(0)=0$ and $f(1)=1$; and. (2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.

Find the units digit of the decimal expansion of \[(15 \plus{} \sqrt{220})^{19} \plus{} (15 \plus{} \sqrt{220})^{82}.\] $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ \text{none of these}$

Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that \[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\] Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that \[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]

Find all positive integers $n$ such that $3^{n}-1$ is divisible by $2^n$.

Determine all pairs of natural numbers $(x; n)$ that satisfy the equation \[x^{3}+2x+1 = 2^{n}.\]

Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.

Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that [b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$; [b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.

Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions: (1) $ f(mn) \equal{} f(m)\plus{}f(n)$, (2) $ f(2008) \equal{} 0$, and (3) $ f(n) \equal{} 0$ for all $ n \equiv 39\pmod {2008}$.

Find the greatest common divisor of the numbers \[ 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.\]

Let $F(z)=\frac{z+i}{z-i}$ for all complex numbers $z\not= i,$ and let $z_n=F(z_{n-1})$ for all positive integers $n.$ Given that $z_0=\frac 1{137}+i$ and $z_{2002}=a+bi,$ where $a$ and $b$ are real numbers, find $a+b.$