15 of the cells of a chessboard 8x8 are chosen. We draw the segments which unite the centers of every two of the chosen squares. Prove that among these segments there are four segments which have the same length.

Let $x,y,z$ be positive real numbers.Prove that $(xy^2+yz^2+zx^2)(x^2y+y^2z+z^2x)(xy+yz+zx)\geq 3(x+y+z)^2(xyz)^2.$

Let ABC be a triangle with angle ACB=60. Let AA' and BB' be altitudes and let T be centroid of the triangle ABC. If A'T and B'T intersect triangle's circumcircle in points M and N respectively prove that MN=AB.

Positive integer $q$ is the $k{}$-successor of positive integer $n{}$ if there exists a positive integer $p{}$ such that $n+p^2=q^2$. Let $A{}$ be the set of all positive integers $n{}$ that have at least a $k{}$-successor, but every $k{}$-successor does not have $k{}$-successors of its own. Prove that $$A=\{7,12\}\cup\{8m+3\mid m\in\mathbb{N}\}\cup\{16m+4\mid m\in\mathbb{N}\}.$$