CL-MO 2012

What is the minimum number of movements that a horse must carry out on chess, on an $8\times 8$ board, to reach the upper right square starting at the lower left? Remember that the horse moves in the usual $L$ -shaped manner.

Let $a_1,a_2,...,a_n$ be all positive integers with $2012$ digits or less, none of which is a $9$ . Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ... +\frac{1}{a_{n}}\le 80.$$

A person enters the social network facebook. He befriends at least one person a day for the first $30$ days. At the end of those $30$ days, it has been exactly $45$ friends. Prove that there is a sequence of consecutive days where made exactly $14$ friends.