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China TST 1987 sequence challenge

Source: China TST 1987, problem 3

May 16, 2005
inductionalgebra unsolvedalgebra

Problem Statement

Let r1=2r_1=2 and rn=∏k=1n−1ri+1r_n = \prod^{n-1}_{k=1} r_i + 1, n≥2.n \geq 2. Prove that among all sets of positive integers such that ∑k=1n1ai<1,\sum^{n}_{k=1} \frac{1}{a_i} < 1, the partial sequences r1,r2,...,rnr_1,r_2, ... , r_n are the one that gets nearer to 1.
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