Define the sequence $a_1, a_2, a_3, \ldots$ by $a_1 = 1$ and, for $n > 1$, \[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\] Prove that there are infinitely many $n$ such that $a_n \equiv n \pmod{2^{2010}}$.

An integer $x$ is selected at random between 1 and $2011!$ inclusive. The probability that $x^x - 1$ is divisible by $2011$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$. [i]Author: Alex Zhu[/i]

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]

Determine all pairs of positive integers $ (m,n)$ such that $ (1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})$ is divisible by $ (1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})$.

Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides \[\binom{p^n}{p}-p^{n-1}. \]

$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$

We call a sequence of $m$ consecutive integers a [i]friendly[/i] sequence if its first term is divisible by $1$, the second by $2$, ..., the $(m-1)^{th}$ by $m-1$, and in addition, the last term is divisible by $m^2$ Does a friendly sequence exist for (a) $m=20$ and (b) $m=11$?

Let $a,b,c,d$ be distinct digits such that the product of the $2$-digit numbers $\overline{ab}$ and $\overline{cb}$ is of the form $\overline{ddd}$. Find all possible values of $a+b+c+d$.

Consider $p$ a prime number and $p$ consecutive positive integers $m_{1}, m_{2}, \ldots, m_{p}$. Choose a permutation $\sigma$ of $1, 2, \ldots, p$. Show that there exist two different numbers $k,l \in \{1,2, \ldots, p\}$ such that $m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)}$ is divisible by $p$.

There is a frog in every vertex of a regular 2n-gon with circumcircle($n \geq 2$). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as $\textsl{a way of jump}$. It turns out that there is $\textsl{a way of jump}$ with respect to 2n-gon, such that the line connecting any two distinct vertice having frogs on it after the jump, does not pass through the circumcentre of the 2n-gon. Find all possible values of $n$.

Find all natural numbers $ n $ for which there exists two natural numbers $ a,b $ such that $$ n=S(a)=S(b)=S(a+b) , $$ where $ S(k) $ denotes the sum of the digits of $ k $ in base $ 10, $ for any natural number $ k. $ [i]Vasile Zidaru[/i] and [i]Mircea Lascu[/i]

There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,2,3,\cdots,n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that (1) for each $k$, the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i+n$ is seat $i$); (2) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break. Find the number of possible values of $n$ with $1<n<1000$.

Which of the following does not divide the number of ordered pairs $(x,y)$ of integers satisfying the equation $x^3 - 13y^3 = 1453$? $ \textbf{a)}\ 2 \qquad\textbf{b)}\ 3 \qquad\textbf{c)}\ 5 \qquad\textbf{d)}\ 7 \qquad\textbf{e)}\ \text{None of above} $

What is the smallest positive integer $n$ such that $2013^n$ ends in $001$ (i.e. the rightmost three digits of $2013^n$ are $001$?

Define a sequence of integers by $T_1 = 2$ and for $n\ge2$, $T_n = 2^{T_{n-1}}$. Find the remainder when $T_1 + T_2 + \cdots + T_{256}$ is divided by 255. [i]Ray Li.[/i]

Solve the equation \[ x^3+2y^3-4x-5y+z^2=2012, \] in the set of integers.

Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

Let $\varphi(k)$ denote the numbers of positive integers less than or equal to $k$ and relatively prime to $k$. Prove that for some positive integer $n$, \[ \varphi(2n-1) + \varphi(2n+1) < \frac{1}{1000} \varphi(2n). \][i]Proposed by Evan Chen[/i]

Does there exist an infinite number of sets $C$ consisting of $1983$ consecutive natural numbers such that each of the numbers is divisible by some number of the form $a^{1983}$, with $a \in \mathbb N, a \neq 1?$

Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.

Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$. Prove that $2011$ divides $a_{2010} + 1$.