Given is an isosceles triangle $ABC$ with $CA=CB$ and angle bisector $BD$, $D \in AC$. The line through the center $O$ of $(ABC)$, perpendicular to $BD$, meets $BC$ at $E$. The line through $E$, parallel to $BD$, meets $AC$ at $F$. Prove that $CE=DF$.

Let $a, b, c \in [0;1]$ be reals such that $ab+bc+ca=1$. Find the minimal and maximal value of $a^3+b^3+c^3$.

Find all triples $(k, m, n)$ of positive integers such that $m$ is a prime and: (1) $kn$ is a perfect square; (2) $\frac{k(k-1)}{2}+n$ is a fourth power of a prime; (3) $k-m^2=p$ where $p$ is a prime; (4) $\frac{n+2}{m^2}=p^4$.