Find all pairs of positive integers $a, b$ such that $\sqrt{a+2\sqrt{b}}=\sqrt{a-2\sqrt{b}}+\sqrt{b}$ .

Let $ABC$ be a triangle with centroid $T$. Denote by $M$ the midpoint of $BC$. Let $D$ be a point on the ray opposite to the ray $BA$ such that $AB = BD$. Similarly, let $E$ be a point on the ray opposite to the ray $CA$ such that $AC = CE$. The segments $T D$ and $T E$ intersect the side $BC$ in $P$ and $Q$, respectively. Show that the points $P, Q$ and $M$ split the segment $BC$ into four parts of equal length.

Determine all positive integers $n$ such that it is possible to fill the $n \times n$ table with numbers $1, 2$ and $-3$ so that the sum of the numbers in each row and each column is $0$.

Let $k$ be a circle with diameter $AB$. A point $C$ is chosen inside the segment $AB$ and a point $D$ is chosen on $k$ such that $BCD$ is an acute-angled triangle, with circumcentre denoted by $O$. Let $E$ be the intersection of the circle $k$ and the line $BO$ (different from $B$). Show that the triangles $BCD$ and $ECA$ are similar.

Given is a group in which everyone has exactly $d$ friends and every two strangers have exactly one common friend. Prove that there are at most $d^2 + 1$ people in this group.

Rational numbers $a, b$ are such that $a+b$ and $a^2+b^2$ are integers. Prove that $a, b$ are integers.

The chess piece sick rook can move along rows and columns as a regular rook, but at most by $2$ fields. We can place sick rooks on a square board in such a way that no two of them attack each other and no field is attacked by more than one sick rook. a) Prove that on $30\times 30$ board, we cannot place more than $100$ sick rooks. b) Find the maximum number of sick rooks which can be placed on $8\times 8$ board. c) Prove that on $32\times 32$ board, we cannot place more than $120$ sick rooks.

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals, such that $\angle BAC = \angle ADB$, $\angle CBD = \angle DCA$, $AB = 15$, $CD = 8$. Show that $ABCD$ is cyclic and find the distance between its circumcenter and the intersection point of its diagonals.

Determine all possible values of the expression $xy+yz+zx$ with real numbers $x, y, z$ satisfying the conditions $x^2-yz = y^2-zx = z^2-xy = 2$.

Let $A_1A_2 ...A_{360}$ be a regular $360$-gon with centre $S$. For each of the triangles $A_1A_{50}A_{68}$ and $A_1A_{50}A_{69}$ determine, whether its images under some $120$ rotations with centre $S$ can have (as triangles) all the $360$ points $A_1, A_2, ..., A_{360}$ as vertices.

Given is a cyclic quadrilateral $ABCD$. Points $K, L, M, N$ lying on sides $AB, BC, CD, DA$, respectively, satisfy $\angle ADK=\angle BCK$, $\angle BAL=\angle CDL$, $\angle CBM =\angle DAM$, $\angle DCN =\angle ABN$. Prove that lines $KM$ and $LN$ are perpendicular.