Let $n \ge 4$ be a positive integer. Into every square of an $n \times n$ array we inscribe an integer so that the sum of the integers inside any $3 \times 3$ square is negative. Find all positive integers $n$ for which this can be done in such a way that the sum of all the numbers in the array is positive.
The base $AB$ of the trapezoid $ABCD$ is longer than the base $CD$, and $\angle ADC$ is a right angle. The diagonals $AC$ and $BD$ are perpendicular. Let $E$ be the foot of the altitude from $D$ to the line $BC$. Prove that $\frac{|AE|}{|BE|} = \frac{|AC| \cdot |CD|}{|AC|^2 - |CD|^2}$.
Let $k$ be a positive integer greater than $1$. Prove: for every non-negative integer $m$ there exist $k$ positive integers $n_1, n_2, \dots, n_k$, such that $$ n_1^2 + n_2^2 + \dots + n_k^2 = 5^{m+k}. $$
Find all functions $g: \mathbb{R} \to \mathbb{R}$, for which there exists a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$, such that $$ f(x + y) = f(x)g(y) + f(y) $$ for all real $x$ and $y$.
Let $n$ be a positive integer. $n \ge 3$. There are $n$ pairwise different numbers written on a blackboard. Show that we can choose two of those numbers so that no number from the blackboard multiplied by $3$ is equal to a multiple of their sum.
The incircle of an acute triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $A_1$, $B_1$ and $C_1$. Let $K_A$, $K_B$ and $K_C$ be the incircles of the triangles $AB_1C_1$, $A_1BC_1$ and $A_1B_1C$.
Let $t_A$ denote the common tangent of $K_B$ and $K_C$ which intersects the segments $AB$ and $AC$ but not the segment $BC$. Let $t_B$ be the common tangent of $K_A$ and $K_C$ which intersects the segments $AB$ and $BC$ but not the segment $AC$ and let $t_C$ be the common tangent of $K_A$ and $K_B$ which intersects the segments $AC$ and $BC$ but not the segment $AB$.
Prove that the lines $t_A$, $t_B$ and $t_C$ intersect at a single point.
There were 64 contestants at a chess tournament. Every pair played a game that ended either with one of them winning or in a draw. If a game ended in a draw, then each of the remaining 62 contestants won against at least one of these two contestants. There were at least two games ending in a draw at the tournament. Show that we can line up all the contestants so that each of them has won against the one standing right behind him.
Find all real numbers $x, y$ and $z$ that solve the system of equations $$
\begin{aligned}
x + y + 2z &= 0 \\
x y - z^{2} &= 0 \\
y^{2} + 5z + 6 &= 0
\end{aligned}
$$
Let $E$ and $F$ be points on the sides $AB$ and $AD$ of a convex quadrilateral $ABCD$ such that $EF \parallel BD$. The segment $CE$ meets the diagonal $BD$ at the point $G$, and the segment $CF$ meets the diagonal $BD$ at the point $H$. Prove: if $AGCH$ is a parallelogram, then $ABCD$ is also a parallelogram.
Given an acute triangle $ABC$ in which $|AB| \gt |BC| \gt |AC|$. Let $D$ be a point on the segment $BC$, different from $C$, such that $|AC| = |AD|$. Denote by $H$ the orthocentre of the triangle $ABC$ and by $A_1$ and $B_1$ the feet of the altitudes from $A$ and $B$. The lines $A_1B_1$ and $DH$ intersect at the point $E$. Prove that the points $B$, $D$, $B_1$ and $E$ are concyclic.
Let $p$ be a prime number and let $a$, $b$ and $c$ be integers divisible by $p$, such that the polynomial $$ q(x) = x^{3} + a x^{2} + b x + c $$ has at least two different integer roots. Prove that $p^{2}$ divides $b$ and $p^{3}$ divides $c$.
Given an acute triangle $ABC$ in which $|AB| \gt |BC| \gt |AC|$. Let $D$ be a point on the segment $BC$, different from $C$, such that $|AC| = |AD|$. Denote by $H$ the orthocentre of the triangle $ABC$ and by $A_{1}$ and $B_{1}$ the feet of the altitudes from $A$ and $B$. The line $DH$ meets the line $AC$ at the point $E$ and the line $A_{1}B_{1}$ at the point $F$. Let $G$ be the intersection of the lines $AF$ and $BH$. Prove that the triangles $HBD$ and $HGE$ are similar.
A $5 \times 7$ table has one red cell and all other cells white. On this table we place tiles of the shape (a tile may be reflected or rotated) so that they lie entirely on the table. Tiles may overlap on the red cell, but not on white cells. Determine all possible positions of the red cell for which the table can be completely covered.
An $8 \times 8$ table is divided into $64$ unit squares. Determine the smallest number of colors needed to color the squares of the table so that every four squares forming a figure of the shape (the figure may be reflected or rotated) are colored with different colors.
(a tile may be reflected or rotated) so that they lie entirely on the table. Tiles may overlap on the red cell, but not on white cells. Determine all possible positions of the red cell for which the table can be completely covered.
(the figure may be reflected or rotated) are colored with different colors.