IMC 2008

Let $n, k$ be positive integers and suppose that the polynomial $x^{2k}-x^k+1$ divides $x^{2n}+x^n+1$ . Prove that $x^{2k}+x^k+1$ divides $x^{2n}+x^n+1$ .

Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.

Let $n$ be a positive integer. Prove that $2^{n-1}$ divides $$\sum_{0\leq k < n/2} \binom{n}{2k+1}5^k.$$