Does there exist a function $f\colon \mathbb{R}\to\mathbb{R}$ such that for any $x>y,$ it satisfies $f(x)-f(y)>\sqrt{x-y}.$

Let a hexagone with a diameter $D$ be given and let $d>\frac D 2.$ On each side of the hexagon one constructs a isosceles triangle with two equal sides of length $d$. Prove that the sum of the areas of those isoscele triangles is greater than the area of a rhombus with side lengths $d$ and a diagonal of length $D$. (The diameter of a polygon is the maximum of the lengths of all its sides and diagonals.)

Find all integers $a, m, n, k,$ such that $(a^m+1)(a^n-1)=15^k.$

What is the maximal number of elements we can choose form the set $\{1, 2, \ldots, 31\}$, such that the sum of any two of them is not a perfect square.

A quadrilateral $ABCD$ is such that $\angle A= \angle C=60^o$ and $\angle B=100^o$. Let $O_1$ and $O_2$ be the centers of the incircles of triangles $ABD$ and $CBD$ respectively. Find the angle between the lines $AO_2$ and $CO_1$.

Find the smallest $n$ such that in an $8\times 8$ chessboard any $n$ cells contain two cells which are at least $3$ knight moves apart from each other.