In a square $ABCD$ let $F$ be the midpoint of $\left[ CD\right] $ and let $E$ be a point on $\left[ AB\right] $ such that $AE>EB$ . the parallel with $\left( DE\right) $ passing by $F$ meets the segment $\left[ BC\right] $ at $H$. Prove that the line $\left( EH\right) $ is tangent to the circle circumscribed with $ABCD$

Find all the positive integers $x,y,z$ satisfiing : $x^{2}+y^{2}+z^{2}=2xyz$

Consider $n$ points $A_1, A_2, \ldots, A_n$ on a circle. How many ways are there if we want to color these points by $p$ colors, so that each two neighbors points are colored with two different colors?

$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that \[bc<2a^2<4bc\]