Let $ n$ be an odd positive integer. Prove that $((n-1)^n+1)^2$ divides $ n(n-1)^{(n-1)^n+1}+n$.

Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.

Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

Let $A$ be a set of positive integers. $A$ is called "balanced" if [and only if] the number of 3-element subsets of $A$ whose elements add up to a multiple of $3$ is equal to the number of 3-element subsets of $A$ whose elements add up to not a multiple of $3$. a. Find a 9-element balanced set. b. Prove that no set of $2013$ elements can be balanced.

$p$ 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]

Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Lewis Chen[/i]

The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{n}= 1+2^{2}+3^{3}+\cdots+n^{n}.\] Prove that there are infinitely many $n$ such that $a_{n}$ is composite.

Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules: i) Any incantation can appear no more than once; ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it; iii)The first person who cannot spell an incantation loses the contest. Answer the following questions: a) If A says '$STAGEPREIMO$' first, then who will win? b) Let $M$ be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are $2009$ and containing only four letters $A,B,C,D$, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in $M$ such that A has a winning strategy by starting with it.

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

Prove that for $n\geq 2$, \[\underbrace{2^{2^{\cdots^{2}}}}_{n\text{ terms}}\equiv \underbrace{2^{2^{\cdots^{2}}}}_{n-1\text{ terms}}\; \pmod{n}.\]

Find all positive integer solutions of equation $n^3 - 2 = k! .$

Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer.

A student had $18$ papers. He seleced some of these papers, then he cut each of them in $18$ pieces.He took these pieces and selected some of them, which he again cut in $18$ pieces each.The student took this procedure untill he got tired .After a time he counted the pieces and got $2012$ pieces .Prove that the student was wrong during the counting.

Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions: (i) $a,b\in\{1,2,\ldots,2013\}$; (ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.

Find all prime numbers $ p$ and $ q$ such that $ p^3 \minus{} q^5 \equal{} (p \plus{} q)^2$.

Let $k$ be a nonzero natural number and $m$ an odd natural number . Prove that there exist a natural number $n$ such that the number $m^n+n^m$ has at least $k$ distinct prime factors.

Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

Solve the equation $p^2-pq-q^3=1$ in prime numbers. [i]A. Golovanov[/i]

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

The number of pairs of integers $ (m,n)$ satisfying the equation \[ m^3 \plus{} 6m^2 \plus{} 5m \equal{} 27n^3 \plus{} 9n^2 \plus{} 9n \plus{} 1\] is $ \text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \text{Infinitely many}$

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

An infinite number of lilypads grow in a line, numbered $\dots$, $-2$, $-1$, $0$, $1$, $2$, $\dots$ Thumbelina and her pet frog start on one of the lilypads. She wants to make a sequence of jumps that will end on either pad $0$ or pad $96$. On each jump, Thumbelina tells her frog the distance (number of pads) to leap, but the frog chooses whether to jump left or right. From which starting pads can she always get to pad $0$ or pad $96$, regardless of her frog's decisions?

Let $a_{1}, a_{2}, a_{3}, ... a_{2011}$ be nonnegative reals with sum $\frac{2011}{2}$, prove : $|\prod_{cyc} (a_{n} - a_{n+1})| = |(a_{1} - a_{2})(a_{2} - a_{3})...(a_{2011}-a_{1})| \le \frac{3 \sqrt3}{16}.$

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, $n^p -p$ is not divisible by $q$.