IMO 1992 SL 15

Origin: PRK

Problem

Does there exist a set M with the following properties? (i) The set M consists of 1992 natural numbers. (ii) Every element in M and the sum of any number of elements have the form mk (m, k \inN, k \geq2).

Solution

The result follows from the following lemma by taking n = 1992\cdot1993 and M = {d, 2d, . . . , 1992d}. Lemma. For every n \inN there exists a natural number d such that all the numbers d, 2d, . . . , nd are of the form mk (m, k \inN, k \geq2). Proof. Let p1, p2, . . . , pn be distinct prime numbers. We shall find d in the form d = 2\alpha23\alpha3 \cdot \cdot \cdot n\alphan, where \alphai \geq0 are integers such that kd is a perfect pkth power. It is sufficient to find \alphai, i = 2, 3, . . . , n, such that \alphai \equiv0 (mod pj) if i ̸= j and \alphai \equiv−1 (mod pj) if i = j. But

the existence of such \alphai’s is an immediate consequence of the Chinese remainder theorem.