There is a knight in the left down corner of $2009 \times 2009$ chessboard. The row and the column containing this corner are painted. The knight cannot move into painted cell and after its move new row and column that contains a square with knight become painted. Is it possible to paint all rows and columns of the chessboard?

Bisectors $AA_1$, $BB_1$, and $CC_1$ of triangle $ABC$ meet at point $I$. The perpendicular bisector to $BB_1$ meets $AA_1,CC_1$ at points $A_0,C_0$ respectively. Prove that the circumcircles of triangles $A_0IC_0$ and $ABC$ touch.

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Can the set of lattice points $\{(x, y) \mid x, y \in \mathbb{Z}, 1 \le x, y \le 252, x \neq y\}$ be colored using 10 distinct colors such that for all $a \neq b$, $b \neq c$, the colors of $(a, b)$ and $(b, c)$ are distinct?

Prove that : $\cos36-\sin18=\frac{1}{2}$

Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

Find the function $f(x)$ that satisfies the equation $$f'(x) + x^2f(x) = 0$$ knowing that $f(1) = e$. Graph this function and calculate the tangent of the curve at the point of abscissa $1$.

Let $ABC$ be a triangle and $AA_1$ be the angle bisector of $A$ ($A_1 \in BC$). The point $P$ is on the segment $AA_1$ and $M$ is the midpoint of the side $BC$. The point $Q$ is on the line connecting $P$ and $M$ such that $M$ is the midpoint of $PQ$. Define $D$ and $E$ as the intersections of $BQ$, $AC$, and $CQ$, $AB$. Prove that $CD=BE$.

For a natural number $n$, let $w(n)$ denote the number of (positive) prime divisors of $n$. Find the smallest positive integer $k$ such that $2^{w(n)} \le k \sqrt[4]{ n}$ for each $n \in N$.

Prove that there exist two strictly increasing sequences $a_{n}$ and $b_{n}$ such that $a_{n}(a_{n} +1)$ divides $b_{n}^2 +1$ for every natural $n$.

The [i]liar's guessing game[/i] is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players. At the start of the game $A$ chooses integers $x$ and $N$ with $1 \le x \le N.$ Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wishes. After each question, player $A$ must immediately answer it with [i]yes[/i] or [i]no[/i], but is allowed to lie as many times as she wants; the only restriction is that, among any $k+1$ consecutive answers, at least one answer must be truthful. After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins; otherwise, he loses. Prove that: 1. If $n \ge 2^k,$ then $B$ can guarantee a win. 2. For all sufficiently large $k$, there exists an integer $n \ge (1.99)^k$ such that $B$ cannot guarantee a win. [i]Proposed by David Arthur, Canada[/i]

Let $ABC$ be a triangle with $\angle A < \angle B \le \angle C$, $M$ and $N$ the midpoints of sides $CA$ and $AB$, respectively, and $P$ and $Q$ the projections of $B$ and $C$ on the medians $CN$ and $BM$, respectively. Prove that the quadrilateral $MNPQ$ is cyclic.

a) Determine if there exists a convex hexagon $ABCDEF$ with $$\angle ABD + \angle AED > 180^{\circ},$$ $$\angle BCE + \angle BFE > 180^{\circ},$$ $$\angle CDF + \angle CAF > 180^{\circ}.$$ b) The same question, with additional condition, that diagonals $AD, BE,$ and $CF$ are concurrent.

Find all integers \(n\) such that there exists a concave pentagon which can be dissected into \(n\) congruent triangles.

All sides of the convex pentagon $ ABCDE$ are of equal length, and $ \angle A \equal{} \angle B \equal{} 90^{\circ}$. What is the degree measure of $ \angle E$? $ \textbf{(A)}\ 90 \qquad \textbf{(B)}\ 108 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$

The altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ meet at point $H$. Point $Q$ is the reflection of the midpoint of $AC$ in line $AA_1$, point $P$ is the midpoint of segment $A_1C_1$. Prove that $\angle QPH = 90^o$. (D.Shvetsov)

Let $ a$, $ b$, $ c$ be positive real numbers. Prove that \[ \dfrac{(2a \plus{} b \plus{} c)^2}{2a^2 \plus{} (b \plus{} c)^2} \plus{} \dfrac{(2b \plus{} c \plus{} a)^2}{2b^2 \plus{} (c \plus{} a)^2} \plus{} \dfrac{(2c \plus{} a \plus{} b)^2}{2c^2 \plus{} (a \plus{} b)^2} \le 8. \]

A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.

Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$: \[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]

Let $ABC$ be a right-angled triangle and $CH$ be the altitude from its right angle $C$. Points $O_1$ and $O_2$ are the incenters of triangles $ACH$ and $BCH$ respectively, $P_1$ and $P_2$ are the touching points of their incircles with $AC$ and $BC$. Prove that lines $O_1P_1$ and $O_2P_2$ meet on $AB$.

Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

On an $8 \times 8$ board, $10$ checkers have been placed, each occupying a square. On each square without a token, a number between $0$ and $8$ is written, which is equal to the number of tokens placed on its neighboring squares. Neighboring cells are those that have a side or a vertex in common. Give a distribution of the tiles that makes the sum of the numbers written on the board the greatest possible.

Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?

Solve in positive integers the equation $1005^x + 2011^y = 1006^z$.