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There exists natural K such that sum > 1993/1000

Source: Turkey IMO TST 1993 #3

July 7, 2011
inequalities unsolvedinequalities

Problem Statement

Let (bnb_n) be a sequence such that bn≥0b_n \geq 0 and bn+12≥b1213+⋯+bn2n3b_{n+1}^2 \geq \frac{b_1^2}{1^3}+\cdots+\frac{b_n^2}{n^3} for all n≥1n \geq 1. Prove that there exists a natural number KK such that ∑n=1Kbn+1b1+b2+⋯+bn≥19931000\sum_{n=1}^{K} \frac{b_{n+1}}{b_1+b_2+ \cdots + b_n} \geq \frac{1993}{1000}
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