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Putnam 2001 B6

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February 27, 2012
Putnamlimitcollege contests

Problem Statement

Assume that (an)n≥1(a_n)_{n \ge 1} is an increasing sequence of positive real numbers such that lim⁡ann=0\lim \tfrac{a_n}{n}=0. Must there exist infinitely many positive integers nn such that an−i+an+i<2ana_{n-i}+a_{n+i}<2a_n for i=1,2,⋯ ,n−1i=1,2,\cdots,n-1?
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