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ASU 203 All Soviet Union MO 1974 f(x_1+x_2)\ge f(x_1)+f(x_2)

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July 4, 2019
functionalgebrainequalities

Problem Statement

Given a function f(x)f(x) on the segment 0≤x≤10\le x\le 1. For all x,f(x)≥0,f(1)=1x, f(x)\ge 0, f(1)=1. For all the couples of (x1,x2)(x_1,x_2) such, that all the arguments are in the segment f(x1+x2)≥f(x1)+f(x2).f(x_1+x_2)\ge f(x_1)+f(x_2). a) Prove that for all xx holds f(x)≤2xf(x) \le 2x.
b) Is the inequality f(x)≤1.9xf(x) \le 1.9x valid?
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