Prove that the integer $145^{n} + 3114\cdot 138^{n}$ is divisible by $1981$ if $n=1981$, and that it is not divisible by $1981$ if $n=1980$.

Find the number of $7$-tuples $(n_1,\ldots,n_7)$ of integers such that \[\sum_{i=1}^7 n_i^6=96957.\]

Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that \[ \left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2 \] for any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$. (Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.)

Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by \[ f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 , \] where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $ i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$). a) Prove that $f(x)$ is divisible with $(x-1)$, but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$; b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$. [i]Sugested by Calin Popescu.[/i]

For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

(a) Is $2005^{2004}$ the sum of two perfect squares? (b) Is $2004^{2005}$ the sum of two perfect squares?

Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that \[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]

Prove that $n^3-n-3$ is not a perfect square for any integer $n$. [i]Calvin Deng.[/i]

A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$. [i]AMM Magazine[/i]

The units digit of $3^{1001}7^{1002}13^{1003}$ is $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $

Do there exist positive integers $a$ and $b$ such that $a^n+n^b$ and $b^n+n^a$ are relatively prime for all natural $n$?

[b]a)[/b] Solve the equation $ x^2-x+2\equiv 0\pmod 7. $ [b]b)[/b] Determine the natural numbers $ n\ge 2 $ for which the equation $ x^2-x+2\equiv 0\pmod n $ has an unique solution modulo $ n. $

Does there exist a sequence $a_1,a_2,a_3,\ldots $ of positive integers such that the sum of every $n$ consecutive elements is divisible by $n^2$ for every positive integer $n$?

Let $ f(x)$ be a polynomial with integer coefficients and let $ p$ be a prime. Denote by $ z_1,...,z_{p\minus{}1}$ the $ (p\minus{}1)$th complex roots of unity. Prove that $ f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}$.

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

[b]Problem 2[/b] Determine all pairs $(n, m)$ of positive integers satisfying the equation $$5^n = 6m^2 + 1\ . $$

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

The sequence of integers $\{ x_{n}\}_{n\ge1}$ is defined as follows: \[x_{1}=1, \;\; x_{n+1}=1+{x_{1}}^{2}+\cdots+{x_{n}}^{2}\;(n=1,2,3 \cdots).\] Prove that there are no squares of natural numbers in this sequence except $x_{1}$.

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]

Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

Let $n$ and $m$ be integers such that $n\leq 2007 \leq m$ and $n^n \equiv -1 \equiv m^m \pmod 5$. What is the least possible value of $m-n$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8 $

What is the maximum number of primes that divide both the numbers $n^3+2$ and $(n+1)^3+2$ where $n$ is a positive integer? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None of above} $

The numbers $1, 3, 7$ and $9$ occur in the decimal representation of an integer. Show that permuting the order of digits one can obtain an integer divisible by $7.$

There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?