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f(n)+ f(n+2) \le 2 f(n+1), exists a line with infinitely many points

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1994 p1

February 20, 2020
Functional inequalityfunctionalgebra

Problem Statement

Let f:N→Nf : N \to N be a function which satisfies f(x)+f(x+2)≤2f(x+1)f(x)+ f(x+2) \le 2 f(x+1) for any x∈Nx \in N. Prove that there exists a line in the coordinate plane containing infinitely many points of the form (n,f(n)),n∈N(n, f(n)), n \in N.
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