Find all triplets of nonnegative integers $(x,y,z)$ and $x\leq y$ such that $x^2+y^2=3 \cdot 2016^z+77$

Find all monic polynomials $P,Q$ which are non-constant, have real coefficients and they satisfy $2P(x)=Q(\frac{(x+1)^2}{2})-Q(\frac{(x-1)^2}{2})$ and $P(1)=1$ for all real $x$.

$ABC$ is an acute isosceles triangle $(AB=AC)$ and $CD$ one altitude. Circle $C_2(C,CD)$ meets $AC$ at $K$, $AC$ produced at $Z$ and circle $C_1(B, BD)$ at $E$. $DZ$ meets circle $(C_1)$ at $M$. Show that: a) $\widehat{ZDE}=45^0$ b) Points $E, M, K$ lie on a line. c) $BM//EC$

A square $ABCD$ is divided into $n^2$ equal small (fundamental) squares by drawing lines parallel to its sides.The vertices of the fundamental squares are called vertices of the grid.A rhombus is called nice when: $\bullet$ It is not a square $\bullet$ Its vertices are points of the grid $\bullet$ Its diagonals are parallel to the sides of the square $ABCD$ Find (as a function of $n$) the number of the nice rhombuses ($n$ is a positive integer greater than $2$).