Determine all real numbers $a$ such that the equation: $$x^8+ax^4+1=0$$have four real roots that form an arithmetic progression.

Let $p$ and $n$ be positive integers such that $p$ is prime and $1 + np$ is a perfect square. Prove that the number $n + 1$ can be expressed as the sum of $p$ perfect squares, where some of them can be equal.

Let $M$ and $N$ be points on the side $BC$ of a triangle $ABC$ such that $BM = CN$ ($M$ lies between $B$ and $N$). Points $P$ and $Q$ lie on $AN$ and $AM$ respectively, so that $\angle PMC =\angle MAB$ and $\angle QNB = \angle NAC$. Prove that $\angle QBC = \angle PCB$.

Eight tiles are located on an $8\times 8$ board in such a way that no pair of them are in the same row or in the same column. Prove that, among the distances between each pair of tiles, we can find two of them that are equal (the distance between two tiles is the distance between the centers of the squares in which they are located).