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This is a system of equation of Vietnamese Mathematical Olympiad 2013, the first day. Solve the system of equations $$\begin{cases} \sqrt{\sin^2 x + \dfrac{1}{\sin^2 x}} + \sqrt{\cos^2 y + \dfrac{1}{\cos^2 y}} = \sqrt{\dfrac{20y}{x+y}},\\ \sqrt{\sin^2 y + \dfrac{1}{\sin^2 y}} + \sqrt{\cos^2 x + \dfrac{1}{\cos^2 x}} = \sqrt{\dfrac{20x}{x+y}}. \end{cases}$$

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Hint: add the equations and show that $\sqrt{\sin^2 x + \dfrac{1}{\sin^2 x}} + \sqrt{\cos^2 x + \dfrac{1}{\cos^2 x}}$ is minimized when $\sin^2 x = \cos^2 x = 1/2$. The rest is easy. – whuber Jan 11 '13 at 18:04
Cross-posted at…. – whuber Jan 13 '13 at 0:19

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