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Replace maxima by minima (BxMO2021 P1)

Source: BxMO2021 problem 1

May 3, 2021
BxMOalgebra

Problem Statement

(a) Prove that for all a,b,c,d∈Ra, b, c, d \in \mathbb{R} with a+b+c+d=0a + b + c + d = 0, max⁡(a,b)+max⁡(a,c)+max⁡(a,d)+max⁡(b,c)+max⁡(b,d)+max⁡(c,d)⩾0. \max(a, b) + \max(a, c) + \max(a, d) + \max(b, c) + \max(b, d) + \max(c, d) \geqslant 0. (b) Find the largest non-negative integer kk such that it is possible to replace kk of the six maxima in this inequality by minima in such a way that the inequality still holds for all a,b,c,d∈Ra, b, c, d \in \mathbb{R} with a+b+c+d=0a + b + c + d = 0.
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