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Algebra with quadratic

Source: Greece National Olympiad 2024, Problem 1

February 24, 2024
quadraticsalgebra

Problem Statement

Let a,b,ca, b, c be reals such that some two of them have difference greater than 122\frac{1}{2 \sqrt{2}}. Prove that there exists an integer xx, such that x2−4(a+b+c)x+12(ab+bc+ca)<0.x^2-4(a+b+c)x+12(ab+bc+ca)<0.
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