Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively. a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$. b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy the equality, \[f(x+f(y))=f(x-f(y))+4xf(y)\]for any $x,y\in\mathbb{R}$.

Let $\mathbb{N}$ denote the set of all positive integers. An ordered pair $(a;b)$ of numbers $a,b\in\mathbb{N}$ is called interesting, if for any $n\in\mathbb{N}$ there exists $k\in\mathbb{N}$ such that the number $a^k+b$ is divisible by $2^n$. Find all interesting ordered pairs of numbers.

Find the maximum number of sets which simultaneously satisfy the following conditions: i) any of the sets consists of $4$ elements, ii) any two different sets have exactly $2$ common elements, iii) no two elements are common to all the sets.

Let $n$ be integer, $n>1.$ An element of the set $M=\{ 1,2,3,\ldots,n^2-1\}$ is called good if there exists some element $b$ of $M$ such that $ab-b$ is divisible by $n^2.$ Furthermore, an element $a$ is called very good if $a^2-a$ is divisible by $n^2.$ Let $g$ denote the number of good elements in $M$ and $v$ denote the number of very good elements in $M.$ Prove that \[v^2+v \leq g \leq n^2-n.\]

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)