OME 2005 Problem 6

In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$ . Prove that: a) $0^o \le A \le 60^o$ . b) The height relative to side $a$ is three times the inradius $r$ . c) The distance from the circumcenter to side $a$ is $R - r$ , where $R$ is the circumradius.