The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation: \[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \] for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.

Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and \[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\] for all natural $n \geq 2$. [i]Peter Boyvalenkov[/i]

A sequence of natural numbers $c_1, c_2,\dots$ is called [i]perfect[/i] if every natural number $m$ with $1\le m \le c_1 +\dots+ c_n$ can be represented as $m =\frac{c_1}{a_1}+\frac{c_2}{a_2}+\dots+\frac{c_n}{a_n}$ Given $n$, find the maximum possible value of $c_n$ in a perfect sequence $(c_i)$.

The exact quantity of gas needed for a car to complete a single loop around a track is distributed among $n$ containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.

Given polynomial $P(x) = a_{0}x^{n}+a_{1}x^{n-1}+\dots+a_{n-1}x+a_{n}$. Put $m=\min \{ a_{0}, a_{0}+a_{1}, \dots, a_{0}+a_{1}+\dots+a_{n}\}$. Prove that $P(x) \ge mx^{n}$ for $x \ge 1$. [i]A. Khrabrov [/i]

Let $n \ge 2$ be an integer. There are $n$ houses in a town. All distances between pairs of houses are different. Every house sends a visitor to the house closest to it. Find all possible values of $n$ (with full justification) for which we can design a town with $n$ houses where every house is visited.

For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that \[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]

A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of $ 67$? [asy]size(200); defaultpen(fontsize(10)); label("1", origin); label("3", (2,0)); label("5", (4,0)); label("$\cdots$", (6,0)); label("97", (8,0)); label("99", (10,0)); label("4", (1,-1)); label("8", (3,-1)); label("12", (5,-1)); label("196", (9,-1)); label(rotate(90)*"$\cdots$", (6,-2));[/asy]

In the space there are 8 points that no four of them are in the plane. 17 of the connecting segments are coloured blue and the other segments are to be coloured red. Prove that this colouring will create at least four triangles. Prove also that four cannot be subsituted by five. Remark: Blue triangles are those triangles whose three edges are coloured blue.

Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area.

Suppose that $\displaystyle{{v_1},{v_2},...,{v_d}}$ are unit vectors in $\displaystyle{{{\Bbb R}^d}}$. Prove that there exists a unitary vector $\displaystyle{u}$ such that $\displaystyle{\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}}$ for $\displaystyle{i = 1,2,...,d}$. [b]Note.[/b] Here $\displaystyle{ \cdot }$ denotes the usual scalar product on $\displaystyle{{{\Bbb R}^d}}$. [i]Proposed by Tomasz Tkocz, University of Warwick.[/i]

For a positive integer $n$, let $a_1, a_2, \ldots a_n$ be nonnegative real numbers such that for all real numbers $x_1>x_2>\ldots>x_n>0$ with $x_1+x_2+\ldots+x_n<1$, the inequality $\sum_{k=1}^na_kx_k^3<1$ holds. Show that \[na_1+(n-1)a_2+\ldots+(n-j+1)a_j+\ldots+a_n\leqslant\frac{n^2(n+1)^2}{4}.\]

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

The numbers $1,2,...,1970$ are written on a board. One is allowed to remove $2$ numbers and to write down their difference instead. When repeated often enough, only one number remains. Show that this number is odd.

Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows: \[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\] Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence. Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.

2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

Let $ p \ge 3$ be a prime number. The sequence $ \{a_{n}\}_{n \ge 0}$ is defined by $ a_{n}=n$ for all $ 0 \le n \le p-1$, and $ a_{n}=a_{n-1}+a_{n-p}$ for all $ n \ge p$. Compute $ a_{p^{3}}\; \pmod{p}$.

Let $\, a,b \,$ be odd positive integers. Define the sequence $\, (f_n ) \,$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for $\, n \,$ sufficiently large and determine the eventual value as a function of $\, a \,$ and $\, b$.

Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$, where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$. The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$. The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$. Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$, where $A$ and $B$ are nonempty decreasing and increasing sets ($\mid \cdot \mid$ denotes the number of elements of the set).

The sequence $ \{a_n\}$ of integers is defined by \[ a_1 \equal{} 2, a_2 \equal{} 7 \] and \[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2. \] Prove that $ a_n$ is odd for all $ n > 1.$

There are 2002 towns in a kingdom. Some of the towns are connected by roads in such a manner that, if all roads from one city closed, one can still travel between any two cities. Every year, the kingdom chooses a non-self-intersecting cycle of roads, founds a new town, connects it by roads with each city from the chosen cycle, and closes all the roads from the original cycle. After several years, no non-self-intersecting cycles remained. Prove that at that moment there are at least 2002 towns, exactly one road going out from each of them.

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.

A sequence of integers $a_1,a_2,a_3,\ldots$ is called [i]exact[/i] if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$.

Let $x_1, x_2, ..., x_{2004}$ be a sequence of integer numbers such that $x_{k+3}=x_{k+2}+x_{k}x_{k+1}$, $\forall 1 \le k \le 2001$. Is it possible that more than half of the elements are negative?