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Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of $\frac{x^m y^n}{(1+x^{2m})(1+y^{2n})}$ is:
(1) 1 (2) $\frac{1}{2}$ (3) $\frac{1}{4}$ (4) $\frac{m+n}{6mn}$
By AM-GM: $1 + x^{2m} \geq 2x^m$, $1 + y^{2n} \geq 2y^n$. Maximum $= \frac{1}{4}$.
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