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min and max inequality in positive reals

Source: Mathematics Regional Olympiad of Mexico Southeast 2021 P3

October 22, 2021
minimummaximuminequalities

Problem Statement

Let a,b,ca, b, c positive reals such that a+b+c=1a+b+c=1. Prove that
min⁡{a(1−b),b(1−c),c(1−a)}≤14\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}
max⁡{a(1−b),b(1−c),c(1−a)}≥29\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}
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