In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

Let $S_1$ and $S_2$ be two circumferences, with centers $O_1$ and $O_2$ respectively, and secants on $M$ and $N$. The line $t$ is the common tangent to $S_1$ and $S_2$ closer to $M$. The points $A$ and $B$ are the intersection points of $t$ with $S_1$ and $S_2$, $C$ is the point such that $BC$ is a diameter of $S_2$, and $D$ the intersection point of the line $O_1O_2$ with the perpendicular line to $AM$ through $B$. Show that $M$, $D$ and $C$ are collinear.

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

If $a, b, c$ are positive real numbers, prove that $$\frac{a}{\sqrt{(a + 2b)^3}}+\frac{b}{\sqrt{(b + 2c)^3}} +\frac{c} {\sqrt{(c + 2a)^3}} \ge \frac{1}{\sqrt{a + b + c}}$$ Alexandru Mihalcu

Given an angle $ \angle QBP$ and a point $ L$ outside the angle $ \angle QBP$. Draw a straight line through $ L$ meeting $ BQ$ in $ A$ and $ BP$ in $ C$ such that the triangle $ \triangle ABC$ has a given perimeter.

Let $n$ be a positive integer. In a coordinate grid, a path from $(0,0)$ to $(2n,2n)$ consists of $4n$ consecutive unit steps $(1,0)$ or $(0,1)$. Prove that the number of paths that divide the square with vertices $(0,0),(2n,0),(2n,2n),(0,2n)$ into 2 regions with even areas is $$\frac{{4n \choose 2n} + {2n \choose n}}{2}$$

If $$u=\text{cot} 22^{\circ}30’ \text{ },\text{ } v= \frac{1}{\text{sin} 22^{\circ}30’}$$ prove that $u$ satisfies a quadratic and $v$ a quartic (4th degree) equation with integral coefficients and with leading coefficients $1$.

Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$. [b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$. [b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.

Let $a_1<a_2<\cdots<a_n$ be positive integers. Some colouring of $\mathbb{Z}$ is periodic with period $t$ such that for each $x\in \mathbb{Z}$ exactly one of $x+a_1,x+a_2,\dots,x+a_n$ is coloured. Prove that $n\mid t$. [i]Andrei Radulescu-Banu[/i]

Let $\mathcal{A}$ be the set of finite sequences of positive integers $a_1,a_2,\dots,a_k$ such that $|a_n-a_{n-1}|=a_{n-2}$ for all $3\leqslant n\leqslant k$. If $a_1=a_2=1$, and $k=18$, determine the number of elements of $\mathcal{A}$.

[b]a)[/b] Show that $ n^2+2n+2007 $ is squarefree for any natural number $ n. $ [b]b)[/b] Prove that for any natural number $ k\ge 2 $ there is a nonnegative integer $ m $ such that $ m^2+2m+2k $ is a perfect square.

In $\triangle A B C$, let $D, E$, and $F$ be the midpoints of the sides of the triangle, and let $P, Q,$ and $R$ be the midpoints of the corresponding medians, $AD ,B E,$ and $C F$, respectively, as shown in the figure at the right. Prove that the value of \[\frac{AQ^2 + A R^2 + B P^2 + B R^2 + C P^2+ C Q^2 }{A B^2 + B C^2 + C A^2}\] does not depend on the shape of $\triangle A B C$ and find that value. [asy] defaultpen(linewidth(0.7)+fontsize(10));size(200); pair A=origin, B=(14,0), C=(9,12), D=midpoint(C--B), E=midpoint(C--A), F=midpoint(A--B), R=midpoint(C--F), P=midpoint(D--A), Q=midpoint(E--B); draw(A--B--C--A, linewidth(1)); draw(A--D^^B--E^^C--F); draw(B--R--A--Q--C--P--cycle, dashed); pair point=centroid(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$P$", P, dir(40)*dir(point--P)); label("$Q$", Q, dir(40)*dir(point--Q)); label("$R$", R, dir(40)*dir(point--R)); dot(P^^Q^^R);[/asy]

Given a polynomial $P(x)=x^4+x^3+3x^2-6x+1$. Calculate $P(\alpha^2+\alpha+1)$ where \[ \alpha=\sqrt[3]{\frac{1+\sqrt{5}}{2}}+\sqrt[3]{\frac{1-\sqrt{5}}{2}} \]

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC, CA, AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all three semicircles has radius $t$. Prove that $$\frac{s}{2} < t \le \frac{s}{2} + \left( 1 - \frac{\sqrt3}{2} \right)r.$$

If there are four numbers $(a,b,c,d)$ in four registers of the calculating machine, they turn into $(a-b,b-c,c-d,d-a)$ numbers whenever you press the button. Prove that if not all the initial numbers are equal, machine will obtain at least one number more than $1985$ after some number of the operations.

In a class of $30$ kids are distributed $430 $ apples . Prove that at least two kids will take the same number of apples.

Let $ m$ and $ n$ be positive integers. Prove that $ 6m | (2m \plus{} 3)^n \plus{} 1$ if and only if $ 4m | 3^n \plus{} 1$

The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win? [i]Alexandr Gribalko[/i]

After elections, every parliament member (PM), has his own absolute rating. When the parliament set up, he enters in a group and gets a relative rating. The relative rating is the ratio of its own absolute rating to the sum of all absolute ratings of the PMs in the group. A PM can move from one group to another only if in his new group his relative rating is greater. In a given day, only one PM can change the group. Show that only a finite number of group moves is possible. [i](A rating is positive real number.)[/i]

Prove that there are infinitely many positive integers that cannot be expressed as the sum of a triangular number and a prime number.

Trains $A$ and $B$ are on the same track a distance $100$ miles apart heading towards one another, each at a speed of $50$ miles per hour. A fly starting out at the front of train $A$ flies towards train $B$ at a speed of $75$ miles per hour. Upon reaching train $B$, the fly turns around and flies towards train $A$, again at $75$ miles per hour. The fly continues flying back and forth between the two trains at $75$ miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train $A$ than to train $B$ before getting squashed?

Let $x, y, z$ denote positive real numbers for which $x+y+z = 1$ and $x > yz$, $y > zx$, $z > xy$. Prove that $$\left(\frac{x - yz}{x + yz}\right)^2+ \left(\frac{y - zx}{y + zx}\right)^2+\left(\frac{z - xy}{z + xy}\right)^2< 1.$$

Let $n$ be a positive integer. There are $n$ lamps, each with a switch that changes the lamp from on to off, or from off to on, each time it is pressed. The lamps are initially all off. You are going to press the switches in a series of rounds. In the first round, you are going to press exactly $1$ switch; in the second round, you are going to press exactly $2$ switches; and so on, so that in the $k$th round you are going to press exactly $k$ switches. In each round you will press each switch at most once. Your goal is to finish a round with all of the lamps switched on. Determine for which $n$ you can achieve this goal.

a) Prove that $10201$ is composite in all bases greater than 2. b) Prove that $10101$ is composite in all bases.

It is given the function $f:\left( \mathbb{R} - \{0\} \right) \times \left( \mathbb{R}-\{0\} \right) \to \mathbb{R}$ such that $f(a,b)= \left| \frac{|b-a|}{|ab|}+\frac{b+a}{ab}-1 \right|+ \frac{|b-a|}{|ab|}+ \frac{b+a}{ab}+1$ where $a,b \not=0$. Prove that: \[ f(a,b)=4 \cdot \text{max} \left\{\frac{1}{a},\frac{1}{b},\frac{1}{2} \right\}\]