IMO 1987 LL AUS5

Origin: AUS

Problem

Let there be given three circles K1, K2, K3 with centers O1, O2, O3 respectively, which meet at a common point P. Also, let K1 \capK2 = {P, A}, K2 \capK3 = {P, B}, K3 \capK1 = {P, C}. Given an arbitrary point X on K1, join X to A to meet K2 again in Y , and join X to C to meet K3 again in Z. (a) Show that the points Z, B, Y are collinear. (b) Show that the area of triangle XY Z is less than or equal to 4 times the area of triangle O1O2O3.