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Problem 2 of Finals

Source: IX International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

September 22, 2018
algebrafunctional equationalgebra unsolved

Problem Statement

a) The real number aa and the continuous function f:[a,∞)→[a,∞)f : [a, \infty) \rightarrow [a, \infty) are such that ∣f(x)−f(y)∣<∣x–y∣|f(x)-f(y)| < |x–y| for every two different x,y∈[a,∞)x, y \in [a, \infty). Is it always true that the equation f(x)=xf(x)=x has only one solution in the interval [a,∞)[a, \infty)?
b) The real numbers aa and bb and the continuous function f:[a,b]→[a,b]f : [a, b] \rightarrow [a, b] are such that ∣f(x)−f(y)∣<∣x–y∣|f(x)-f(y)| < |x–y|, for every two different x,y∈[a,b]x, y \in [a, b]. Is it always true that the equation f(x)=xf(x)=x has only one solution in the interval [a,b][a, b]?
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