Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME Prove it United States

Problem:
Let $x$ , $y$ , and $z$ be distinct real numbers that sum to $0$ . Find the maximum possible value of
$\frac{x y + y z + z x}{x^{2} + y^{2} + z^{2}}.$

Solution

Solution:
$-1/2$

Note that $0 = (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x$ . Rearranging, we get that $x y + y z + z x = -\frac{1}{2}(x^2 + y^2 + z^2)$ , so that in fact the quantity is always equal to $-1/2$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.