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R = 2p, p prime, sidelengths wanted in triangle

Source: Romania IMO TST 1992 p5

February 19, 2020
circumradiussidelenghtsgeometry

Problem Statement

Let OO be the circumcenter of an acute triangle ABCABC. Suppose that the circumradius of the triangle is R=2pR = 2p, where pp is a prime number. The lines AO,BO,COAO,BO,CO meet the sides BC,CA,ABBC,CA,AB at A1,B1,C1A_1,B_1,C_1, respectively. Given that the lengths of OA1,OB1,OC1OA_1,OB_1,OC_1 are positive integers, find the side lengths of the triangle.
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