IMO 1982 LL CAN18

Origin: CAN

Problem

You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that (a + ab−1a)−1 + (a + b)−1 = a−1, where x−1 is the element for which x−1x = xx−1 = e, where e is the element of the system such that for all a the equality ea = ae = a holds.