IMO 1979 LL POL53

Origin: POL

Problem

An infinite increasing sequence of positive integers nj (j = 1, 2, . . . ) has the property that for a certain c, N  nj\leqN nj \leqc, for every N > 0 Prove that there exist finitely many sequences m(i) j (i = 1, 2, . . . , k) such that {n1, n2, . . . } = %k i=1{m(i) 1 , m(i) 2 , . . . } and m(i) j+1 > 2m(i) j (1 \leqi \leqk, j = 1, 2, . . . ).