RO-TST 1992 Problem 5

Let $O$ be the circumcenter of an acute triangle $ABC$ . Suppose that the circumradius of the triangle is $R = 2p$ , where $p$ is a prime number. The lines $AO,BO,CO$ meet the sides $BC,CA,AB$ at $A_1,B_1,C_1$ , respectively. Given that the lengths of $OA_1,OB_1,OC_1$ are positive integers, find the side lengths of the triangle.