TUYMAADA 2016

The sequence $(a_n)$ is defined by $a_1=0$ , $$a_{n+1}={a_1+a_2+\ldots+a_n\over n}+1.$$ Prove that $a_{2016}>{1\over 2}+a_{1000}$ .

Altitudes $AA_1$ , $BB_1$ , $CC_1$ of an acute triangle $ABC$ meet at $H$ . $A_0$ , $B_0$ , $C_0$ are the midpoints of $BC$ , $CA$ , $AB$ respectively. Points $A_2$ , $B_2$ , $C_2$ on the segments $AH$ , $BH$ , $HC_1$ respectively are such that $\angle A_0B_2A_2 = \angle B_0C_2B_2 = \angle C_0A_2C_2 =90^\circ$ . Prove that the lines $AC_2$ , $BA_2$ , $CB_2$ are concurrent.

The ratio of prime numbers $p$ and $q$ does not exceed 2 ( $p\ne q$ ). Prove that there are two consecutive positive integers such that the largest prime divisor of one of them is $p$ and that of the other is $q$ .