In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$. Proposed by Fatemeh Sajadi

P(x) is polynomial such that, polynomial P(P(x)) is strictly monotone in all real number line. Prove that polynomial P(x) is also strictly monotone in all real number line.

Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$. Proposed by Viktor Simjanoski, Macedonia

A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.