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If x,y are positive real numbers and m, n are positive integers, then prove that `(x^(n) y^(m))/((1 + x^(2n))(1 + y^(2m))) le (1)/(4)`

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To prove that \[ \frac{x^n y^m}{(1 + x^{2n})(1 + y^{2m})} \leq \frac{1}{4} \] for positive real numbers \(x, y\) and positive integers \(m, n\), we will use the AM-GM (Arithmetic Mean-Geometric Mean) inequality. ...
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