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Bosnia and Herzegovina TST 1999 Day 1 Problem 3

Source: Bosnia and Herzegovina Team Selection Test 1999

September 20, 2018
functionalgebraInjective

Problem Statement

Let f:[0,1]→Rf : [0,1] \rightarrow \mathbb{R} be injective function such that f(0)+f(1)=1f(0)+f(1)=1. Prove that exists x1x_1, x2∈[0,1]x_2 \in [0,1], x1≠x2x_1 \neq x_2 such that 2f(x1)<f(x2)+122f(x_1)<f(x_2)+\frac{1}{2}. After that state at least one generalization of this result
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