is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

How many integer pairs $(x,y)$ are there such that \[0\leq x < 165, \quad 0\leq y < 165 \text{ and } y^2\equiv x^3+x \pmod {165}?\] $ \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 99 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 315 \qquad\textbf{(E)}\ \text{None of above} $

Show that \[ \prod_{1\leq x < y \leq \frac{p\minus{}1}{2}} (x^2\plus{}y^2) \equiv (\minus{}1)^{\lfloor\frac{p\plus{}1}{8}\rfloor} \;(\textbf{mod}\;p\ ) \] for every prime $ p\equiv 3 \;(\textbf{mod}\;4\ )$. [J. Suranyi]

In our number system the base is ten. If the base were changed to four you would count as follows: $ 1,2,3,10,11,12,13,20,21,22,23,30,\ldots$ The twentieth number would be: $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 38 \qquad\textbf{(C)}\ 44 \qquad\textbf{(D)}\ 104 \qquad\textbf{(E)}\ 110$

(a) Show that $n(n + 1)(n + 2)$ is divisible by $6$. (b) Show that $1^{2015} + 2^{2015} + 3^{2015} + 4^{2015} + 5^{2015} + 6^{2015}$ is divisible by $7$.

Evaluate \[\sqrt{2013+276\sqrt{2027+278\sqrt{2041+280\sqrt{2055+\ldots}}}}\]

The first two terms of a sequence are $ a_1 \equal{} 1$ and $ a_2 \equal{} \frac {1}{\sqrt3}$. For $ n\ge1$, \[ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} a_{n \plus{} 1}}{1 \minus{} a_na_{n \plus{} 1}}. \]What is $ |a_{2009}|$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 \minus{} \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 \plus{} \sqrt3$

Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

For how many positive integers $n$ less than or equal to $1000$ is \[(\sin t + i \cos t)^n=\sin nt + i \cos nt\] true for all real $t$?

Ingrid and Erik are playing a game. For a given odd prime $p$, the numbers $1, 2, 3, ..., p-1$ are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is $1 \pmod p$ after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out. The player who has received the most points by the end of the game wins. If both players have the same score, the game ends in a draw. For each $p$, determine which player (if any) has a winning strategy

A $100 \times 100$ table is given. At the beginning, every unit square has number $"0"$ written in them. Two players playing a game and the game stops after $200$ steps (each player plays $100$ steps). In every step, one can choose a row or a column and add $1$ to the written number in all of it's squares $\pmod 3.$ First player is the winner if more than half of the squares ($5000$ squares) have the number $"1"$ written in them, Second player is the winner if more than half of the squares ($5000$ squares) have the number $"0"$ written in them. Otherwise, the game is draw. Assume that both players play at their best. What will be the result of the game ? [i]Proposed by Mahyar Sefidgaran[/i]

Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.

Let $L_n$ be the least common multiple of the integers $1,2,\dots,n$. For example, $L_{10} = 2{,}520$ and $L_{30} = 2{,}329{,}089{,}562{,}800$. Find the remainder when $L_{31}$ is divided by $100{,}000$. [i]Proposed by Evan Chen[/i]

Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.

Let $ A\equal{}\{1,2,\ldots, 2008\}$. We will say that set $ X$ is an $ r$-set if $ \emptyset \neq X \subset A$, and $ \sum_{x\in X} x \equiv r \pmod 3$. Let $ X_r$, $ r\in\{0,1,2\}$ be the set of $ r$-sets. Find which one of $ X_r$ has the most elements.

Let $m$ and $n$ be positive integers such that $m>n$ and $m \equiv n \pmod{2}$. If $(m^2-n^2+1) \mid n^2-1$, then prove that $m^2-n^2+1$ is a perfect square. (proposed by G. Batzaya, folklore)

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

Let $\mathbb{N}$ denote the set of all positive integers. An ordered pair $(a;b)$ of numbers $a,b\in\mathbb{N}$ is called [i]interesting[/i], if for any $n\in\mathbb{N}$ there exists $k\in\mathbb{N}$ such that the number $a^k+b$ is divisible by $2^n$. Find all [i]interesting[/i] ordered pairs of numbers.

If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.

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

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

There are some counters in some cells of $100\times 100$ board. Call a cell [i]nice[/i] if there are an even number of counters in adjacent cells. Can exactly one cell be [i]nice[/i]? [i]K. Knop[/i]

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? ${ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 $

Find the last five digits of $1^{100}+2^{100}+3^{100}+...+999999^{100}$