Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

How many one-to-one functions $f : \{1, 2, \cdots, 9\} \rightarrow \{1, 2, \cdots, 9\}$ satisfy (i) and (ii)? (i) $f(1)>f(2)$, $f(9)<9$. (ii) For each $i=3, 4, \cdots, 8$, if $f(1), \cdots, f(i-1)$ are smaller than $f(i)$, then $f(i+1)$ is also smaller than $f(i)$.

For nonnegative integers $m$ and $n$, define the sequence $a(m,n)$ of real numbers as follows. Set $a(0,0)=2$ and for every natural number $n$, set $a(0,n)=1$ and $a(n,0)=2$. Then for $m,n\geq1$, define \[ a(m,n)=a(m-1,n)+a(m,n-1). \] Prove that for every natural number $k$, all the roots of the polynomial $P_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i}$ are real.

The sequence $ u_n$, $ n\equal{}0,1,2,...$ is defined by $ u_0\equal{}0, u_1\equal{}1$ and for each $ n \ge 1$, $ u_{n\plus{}1}$ is the smallest positive integer greater than $ u_n$ such that $ \{ u_0,u_1,...,u_{n\plus{}1} \}$ contains no three elements in arithmetic progression. Find $ u_{100}$.

Let $x,y$ and $z$ be positive real numbers. Show that $x^2+xy^2+xyz^2\ge 4xyz-4$.

Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

Prove that any positive integer can be represented as a sum of Fibonacci numbers, no two of which are consecutive.

Determine if there exists a positive integer $n$ such that $n$ has exactly $2000$ prime divisors and $2^{n}+1$ is divisible by $n$.

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : \[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\] for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.

Given a permutation ($a_0, a_1, \ldots, a_n$) of the sequence $0, 1,\ldots, n$. A transportation of $a_i$ with $a_j$ is called legal if $a_i=0$ for $i>0$, and $a_{i-1}+1=a_j$. The permutation ($a_0, a_1, \ldots, a_n$) is called regular if after a number of legal transportations it becomes ($1,2, \ldots, n,0$). For which numbers $n$ is the permutation ($1, n, n-1, \ldots, 3, 2, 0$) regular?

Suppose $n$ is a natural number. In how many ways can we place numbers $1,2,....,n$ around a circle such that each number is a divisor of the sum of it's two adjacent numbers?

Given a positive integer number $n$, determine the minimum of \[\max \left\{\dfrac{x_1}{1 + x_1},\, \dfrac{x_2}{1 + x_1 + x_2},\, \cdots,\, \dfrac{x_n}{1 + x_1 + x_2 + \cdots + x_n}\right\},\] as $x_1, x_2, \ldots, x_n$ run through all non-negative real numbers which add up to $1$. [i]Kvant Magazine[/i]

The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n \equal{} 0, 1, 2, \ldots$ by the equalities \[ a_0 \equal{} \frac {\sqrt {2}}{2}, \quad a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}} \] and \[ b_0 \equal{} 1, \quad b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n} \] Prove the inequalities for every $ n \equal{} 0, 1, 2, \ldots$ \[ 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n. \]

Let $f : [0, 1] \to [0, 1]$ satisfy $f(0) = 0, f(1) = 1$ and \[f(x + y) - f(x) = f(x) - f(x - y)\] for all $x, y \geq 0$ with $x - y, x + y \in [0, 1].$ Prove that $f(x) = x$ for all $x \in [0, 1].$

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

Let $n,k$ be given positive integers with $n>k$. Prove that: \[ \frac{1}{n+1} \cdot \frac{n^n}{k^k (n-k)^{n-k}} < \frac{n!}{k! (n-k)!} < \frac{n^n}{k^k(n-k)^{n-k}} \]

On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.

Determine all pairs $(a, b)$ of real numbers such that $a\lfloor bn\rfloor =b\lfloor an\rfloor$ for all positive integer $n$.

What is the least number of moves it takes a knight to get from one corner of an $n\times n$ chessboard, where $n\ge 4$, to the diagonally opposite corner?

Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers such that for each $i,j\in \mathbb N$, number of the divisors of $i+j$ and $a_i+a_j$ is equal. (an increasing sequence is a sequence that if $i\le j$, then $a_i\le a_j$.)

Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eight row and eight column) and places the other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest and the least number that appear in the second row from the top? [asy] add(grid(7,7)); label("$\dots$", (0.5,0.5)); label("$\dots$", (1.5,0.5)); label("$\dots$", (2.5,0.5)); label("$\dots$", (3.5,0.5)); label("$\dots$", (4.5,0.5)); label("$\dots$", (5.5,0.5)); label("$\dots$", (6.5,0.5)); label("$\dots$", (1.5,0.5)); label("$\dots$", (0.5,1.5)); label("$\dots$", (0.5,2.5)); label("$\dots$", (0.5,3.5)); label("$\dots$", (0.5,4.5)); label("$\dots$", (0.5,5.5)); label("$\dots$", (0.5,6.5)); label("$\dots$", (6.5,0.5)); label("$\dots$", (6.5,1.5)); label("$\dots$", (6.5,2.5)); label("$\dots$", (6.5,3.5)); label("$\dots$", (6.5,4.5)); label("$\dots$", (6.5,5.5)); label("$\dots$", (0.5,6.5)); label("$\dots$", (1.5,6.5)); label("$\dots$", (2.5,6.5)); label("$\dots$", (3.5,6.5)); label("$\dots$", (4.5,6.5)); label("$\dots$", (5.5,6.5)); label("$\dots$", (6.5,6.5)); label("$17$", (1.5,1.5)); label("$18$", (1.5,2.5)); label("$19$", (1.5,3.5)); label("$20$", (1.5,4.5)); label("$21$", (1.5,5.5)); label("$16$", (2.5,1.5)); label("$5$", (2.5,2.5)); label("$6$", (2.5,3.5)); label("$7$", (2.5,4.5)); label("$22$", (2.5,5.5)); label("$15$", (3.5,1.5)); label("$4$", (3.5,2.5)); label("$1$", (3.5,3.5)); label("$8$", (3.5,4.5)); label("$23$", (3.5,5.5)); label("$14$", (4.5,1.5)); label("$3$", (4.5,2.5)); label("$2$", (4.5,3.5)); label("$9$", (4.5,4.5)); label("$24$", (4.5,5.5)); label("$13$", (5.5,1.5)); label("$12$", (5.5,2.5)); label("$11$", (5.5,3.5)); label("$10$", (5.5,4.5)); label("$25$", (5.5,5.5)); [/asy] $\textbf{(A) }367 \qquad \textbf{(B) }368 \qquad \textbf{(C) }369 \qquad \textbf{(D) }379 \qquad \textbf{(E) }380$

Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and \[\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}\] for every natural number $m > 0$ and for every integer $n$. Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$