CN-SMO 2010

$ABC$ is a triangle with a right angle at $C$ . $M_1$ and $M_2$ are two arbitrary points inside $ABC$ , and $M$ is the midpoint of $M_1M_2$ . The extensions of $BM_1,BM$ and $BM_2$ intersect $AC$ at $N_1,N$ and $N_2$ respectively. Prove that $\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}$ Attachments:

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AB$ at $F$ , intersects the line $AD$ again at $H$ and the line $CF$ again at $K$ . Prove that $\frac{FD\times HK}{FH\times DK}=3$ Attachments:

Let $a$ and $b$ be positive integers such that $1\leq a<b\leq 100$ . If there exists a positive integer $k$ such that $ab|a^k+b^k$ , we say that the pair $(a, b)$ is good. Determine the number of good pairs.