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Four variables ineq

Source: Benelux MO 2019 P1

April 28, 2019
inequalities2019

Problem Statement

[*]Let a,b,c,da,b,c,d be real numbers with 0⩽a,b,c,d⩽10\leqslant a,b,c,d\leqslant 1. Prove that ab(a−b)+bc(b−c)+cd(c−d)+da(d−a)⩽827.ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.[/*] [*]Find all quadruples (a,b,c,d)(a,b,c,d) of real numbers with 0⩽a,b,c,d⩽10\leqslant a,b,c,d\leqslant 1 for which equality holds in the above inequality.
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