A three by three square is filled with positive integers. Each row contains three different integers, the sums of each row are all the same, and the products of each row are all different. What is the smallest possible value for the sum of each row?

Let an acute angled triangle $ABC$ be given. Prove that the circles whose diameters are $AB$ and $AC$ have a point of intersection on $BC$.

There are $16$ competitors in a tournament, all of whom have different playing strengths and in any match between two players the stronger player always wins. Show that it is possible to find the strongest and second strongest players in $18$ matches.

Find all pairs of positive integers $m$ and $n$ such that $$(m + 1)! + (n + 1)! = m^2n.$$

Let a square $ABCD$ with sides of length $1$ be given. A point $X$ on $BC$ is at distance $d$ from $C$, and a point $Y$ on $CD$ is at distance $d$ from $C$. The extensions of: $AB$ and $DX$ meet at $P$, $AD$ and $BY$ meet at $Q, AX$ and $DC$ meet at $R$, and $AY$ and $BC$ meet at $S$. If points $P, Q, R$ and $S$ are collinear, determine $d$.

Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$

Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$.

In triangle $ABC$, the altitude from $B$ is tangent to the circumcircle of $ABC$. Prove that the largest angle of the triangle is between $90^o$ and $135^o$. If the altitudes from both $B$ and from $C$ are tangent to the circumcircle, then what are the angles of the triangle?

Chris and Michael play a game on a board which is a rhombus of side length $n$ (a positive integer) consisting of two equilateral triangles, each of which has been divided into equilateral triangles of side length $ 1$. Each has a single token, initially on the leftmost and rightmost squares of the board, called the “home” squares (the illustration shows the case $n = 4$). A move consists of moving your token to an adjacent triangle (two triangles are adjacent only if they share a side). To win the game, you must either capture your opponent’s token (by moving to the triangle it occupies), or move on to your opponent’s home square. Supposing that Chris moves first, which, if any, player has a winning strategy?

Let a point $P$ inside a parallelogram $ABCD$ be given such that $\angle APB +\angle CPD = 180^o$. Prove that $AB \cdot AD = BP \cdot DP + AP \cdot CP$.

Prove that for any three distinct positive real numbers $a, b$ and $c$: $$\frac{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}{(a - b)^3 + (b - c)^3 + (c - a)^3}> 8abc.$$

Consider the set $G$ of $2011^2$ points $(x, y)$ in the plane where $x$ and $y$ are both integers between $ 1$ and $2011$ inclusive. Let $A$ be any subset of $G$ containing at least $4\times 2011\times \sqrt{2011}$ points. Show that there are at least $2011^2$ parallelograms whose vertices lie in $A$ and all of whose diagonals meet at a single point.