In the triangle $\vartriangle ABC$ we have $| AB |^3 = | AC |^3 + | BC |^3$. Prove that $\angle C> 60^o$ .

Prove that for every acute angle $\alpha$, $\sin (\cos \alpha) < \cos(\sin \alpha)$.

Write down $f(n)$ for the greatest odd divisor of $n \in N_0$. (a) Determine $f (n + 1) + f (n + 2) + ... + f(2n)$. (b) Determine $f(1) + f(2) + f(3) + ... + f(2n)$.

Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.