IMO 1970 LL GDR28

Origin: GDR

Problem

A set G with elements u, v, w, . . . is a group if the following conditions are fulfilled: (1) There is a binary algebraic operation ◦defined on G such that for all u, v \inG there is a w \inG with u ◦v = w. (2) This operation is associative; i.e., for all u, v, w \inG, (u ◦v) ◦w = u ◦(v ◦w). (3) For any two elements u, v \inG there exists an element x \inG such that u ◦x = v, and an element y \inG such that y ◦u = v. Let K be a set of all real numbers greater than 1. On K is defined an operation by a ◦b = ab + (a2 −1)(b2 −1). Prove that K is a group.