IMO 1997 SL 6

Origin: IRE

Problem

(a) Let n be a positive integer. Prove that there exist distinct positive integers x, y, z such that xn−1 + yn = zn+1. (b) Let a, b, c be positive integers such that a and b are relatively prime and c is relatively prime either to a or to b. Prove that there exist infinitely many triples (x, y, z) of distinct positive integers x, y, z such that xa + yb = zc. Original formulation: Let a, b, c, n be positive integers such that n is odd and ac is relatively prime to 2b. Prove that there exist distinct positive integers x, y, z such that (i) xa + yb = zc, and (ii) xyz is relatively prime to n.

Solution

(a) One solution is x = 2n23n+1, y = 2n2−n3n, z = 2n2−2n+23n−1. (b) Suppose w.l.o.g. that gcd(c, a) = 1. We look for a solution of the form x = pm, y = pn, z = qpr, p, q, m, n, r \inN. Then xa +yb = pma +pnb and zc = qcprc, and we see that it is enough to assume ma −1 = nb = rc (there are infinitely many such triples (m, n, r)) and qc = p + 1.