IMO 1972 LL GBR19

Origin: GBR

Problem

Let S be a subset of the real numbers with the following properties: (i) If x \inS and y \inS, then x −y \inS; (ii) If x \inS and y \inS, then xy \inS; (iii) S contains an exceptional number x′ such that there is no number y in S satisfying x′y + x′ + y = 0; (iv) If x \inS and x ̸= x′, there is a number y in S such that xy+x+y = 0. Show that (a) S has more than one number in it; (b) x′ ̸= −1 leads to a contradiction; (c) x \inS and x ̸= 0 implies 1/x \inS.