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10th EMC - Balanced Quadruples

Source: 10th European Mathematical Cup - Problem J1

December 22, 2021
emcEuropean Mathematical CupInequalityalgebra

Problem Statement

We say that a quadruple of nonnegative real numbers (a,b,c,d)(a,b,c,d) is balanced if a+b+c+d=a2+b2+c2+d2.a+b+c+d=a^2+b^2+c^2+d^2. Find all positive real numbers xx such that (x−a)(x−b)(x−c)(x−d)≥0(x-a)(x-b)(x-c)(x-d)\geq 0 for every balanced quadruple (a,b,c,d)(a,b,c,d). \\ \\ (Ivan Novak)
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