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f((x+y)/2) < (f(x)+ f(y))/2 , a_n = f(n) does not contain arithmetic progression

Source: Romania IMO TST 1993 2.1

February 17, 2020
arithmetic sequencefunctionIncreasingalgebra

Problem Statement

Let f:R+→Rf : R^+ \to R be a strictly increasing function such that f(x+y2)<f(x)+f(y)2f\left(\frac{x+y}{2}\right) < \frac{f(x)+ f(y)}{2} for all x,y>0x,y > 0. Prove that the sequence an=f(n)a_n = f(n) (n∈Nn \in N) does not contain an infinite arithmetic progression.
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