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f(n+1) >= f(n) >= n/(n=1) * f(2n)

Source: China TST 1996, problem 2

May 17, 2005
functionlimitinductionstrong inductionalgebra unsolvedalgebra

Problem Statement

SS is the set of functions f:N→Rf:\mathbb{N} \to \mathbb{R} that satisfy the following conditions: I. f(1)=2f(1) = 2 II. f(n+1)≥f(n)≥nn+1f(2n)f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n) for n=1,2,…n = 1, 2, \ldots Find the smallest M∈NM \in \mathbb{N} such that for any f∈Sf \in S and any n∈N,f(n)<Mn \in \mathbb{N}, f(n) < M.
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