If you like geometry I recommend problem 2 from that 1987 IMO. Simple formulation, elegant solution. Hard, but not crazy hard imo.
"In an acute-angled triangle ABC the interior bisector of the angle A intersects BC at L and intersects the circumcircle of ABC again at N. From point L perpendiculars are drawn to AB and AC, the feet of these perpendiculars being K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC have equal areas."
