For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.

[b]numbers $n^2+1$[/b] Prove that there are infinitely many natural numbers of the form $n^2+1$ such that they don't have any divisor of the form $k^2+1$ except $1$ and themselves. time allowed for this question was 45 minutes.

In every $1\times1$ square of an $m\times n$ table we have drawn one of two diagonals. Prove that there is a path including these diagonals either from left side to the right side, or from the upper side to the lower side.

We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set \[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero). [i]Proposed by Javad Abedi[/i]

Let $(x_n)_{n\ge 1}$ a sequence defined by $x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*$. Prove that $\lim_{n\to \infty} x_n=1$ and evaluate $\lim_{n\to \infty} x_n^n$.

Let $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$, and suppose that the number $a$ satisfies the equation $a=\cos a$. Express $f'(a)$ as a polynomial in $a$.

Evaluate the following : \[ \int_{0}^{\infty}\left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots \right)\;\left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2\cdot 6^2}+\cdots \right)\,\mathrm{d}x \]

One hundred pirates played cards. When the game was over, each pirate calculated the amount he won or lost. The pirates have a gold sand as a currency; each has enough to pay his debt. Gold could only change hands in the following way. Either one pirate pays an equal amount to every other pirate, or one pirate receives the same amount from every other pirate. Prove that after several such steps, it is possible for each winner to receive exactly what he has won and for each loser to pay exactly what he has lost. [i](4 points)[/i]

Let $S$ denote the set of rational numbers in the interval $(0,1)$. Determine, with proof, if there exists a subset $T$ of $S$ such that every element in $S$ can be uniquely written as the sum of finitely many distinct elements in $T$.

Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$

Let $c$ be a fixed positive integer, and $\{x_k\}^{\inf}_{k=1}$ be a sequence such that $x_1=c$ and $x_n=x_{n-1}+\lfloor \frac{2x_{n-1}-2}{n} \rfloor$ for $n \ge 2$. Determine the explicit formula of $x_n$ in terms of $n$ and $c$. (Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Suppose $\{a(n) \}_{n=1}^{\infty}$ is a sequence that: \[ a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3\] and $a(1)=a(2)=1$. Prove that for each $n \geq 1$ , $a(2n) \leq 2a(n)$.

Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

Let the $A$ be the set of all nonenagative integers. It is given function such that $f:\mathbb{A}\rightarrow\mathbb{A}$ with $f(1) = 1$ and for every element $n$ od set $A$ following holds: [b]1)[/b] $3 f(n) \cdot f(2n+1) = f(2n) \cdot (1+3 \cdot f(n))$; [b]2)[/b] $f(2n) < 6f(n)$, Find all solutions of $f(k)+f(l) = 293$, $k<l$.

Consider an infinite sequence $a_1,a_2,\cdots$ whose terms all belong to $\left\{1,2\right\}$. A positive integer with $n$ digits is said to be [i]good[/i] if its decimal representation has the form $a_ra_{r+1}\cdots a_{r+(n-1)}$, for some positive integer $r$. Suppose that there are at least $2008$ [i]good[/i] numbers with a million digits. Prove that there are at least $2008$ [i]good[/i] numbers with $2007$ digits.