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Fixed point in cute polynomial

Source: Brazilian Mathematical Olympiad 2024, Level U, Problem 2

October 12, 2024
real analysisderivativeFixed pointalgebrapolynomial

Problem Statement

For each pair of integers j,k≥2 j, k \geq 2 , define the function fjk:R→R f_{jk} : \mathbb{R} \to \mathbb{R} given by
fjk(x)=1−(1−xj)k. f_{jk}(x) = 1 - (1 - x^j)^k.
(a) Prove that for any integers j,k≥2 j, k \geq 2 , there exists a unique real number pjk∈(0,1) p_{jk} \in (0, 1) such that fjk(pjk)=pjk f_{jk}(p_{jk}) = p_{jk} . Furthermore, defining λjk:=fjk′(pjk) \lambda_{jk} := f'_{jk}(p_{jk}) , prove that λjk>1 \lambda_{jk} > 1 .
(b) Prove that pjkj=1−pkj p^j_{jk} = 1 - p_{kj} for any integers j,k≥2 j, k \geq 2 .
(c) Prove that λjk=λkj \lambda_{jk} = \lambda_{kj} for any integers j,k≥2 j, k \geq 2 .
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