2004 JBMO Problems/Problem 1

Problem

Prove that the inequality \[\frac{ x+y}{x^2-xy+y^2 } \leq \frac{ 2\sqrt 2 }{\sqrt{ x^2 +y^2 } }\] holds for all real numbers $x$ and $y$, not both equal to 0.


Solution

Since the inequality is homogeneous, we can assume WLOG that xy = 1.


Now, substituting $m = (x+y)^2$, we have:

$m = x^2 + y^2 + 2xy = x^2 + y^2 + 2$ $\geq 2\sqrt {xy} + 2 = 4$, thus we have $m \geq 4$


Now squaring both sides of the inequality, we get: \[\frac{m}{(m-3)^2 } \leq \frac{8}{m-2}\] after cross multiplication and simplification we get: $7m^2 -46m + 72 \geq 0$ or, $7(m-4)^2 +10(m-4) \geq 0$ which is always true since $m \geq 4$.


$Kris17$

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