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[SOLVED] Prove abcd≥3

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anemone

MHB POTW Director
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Feb 14, 2012
3,642
Let $a,\,b,\,c$ and $d>0$ and $\dfrac{1}{1+a^4}+\dfrac{1}{1+b^4}+\dfrac{1}{1+c^4}+\dfrac{1}{1+d^4}=1$. Prove that $abcd\ge 3$.
 
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anemone

MHB POTW Director
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Feb 14, 2012
3,642
Let $a^2=\tan \alpha,\,b^2=\tan \beta,\,c^2=\tan \gamma,\,d^2=\tan \delta$. Then $\cos^2 \alpha+\cos^2 \beta+\cos^2 \gamma+\cos^2 \delta=1$. By the AM-GM inequality,

$\sin^2 \alpha=\cos^2 \beta+\cos^2 \gamma+\cos^2 \delta \ge 3(\cos \beta \cos \gamma \cos \delta)^{\small \dfrac{2}{3}}$

Multiplying this and three other similar inequalities, we have

$\sin^2 \alpha \sin^2 \beta \sin^2 \gamma \sin^2 \delta=81\cos^2 \alpha \cos^2 \beta \cos^2 \gamma \cos^2 \delta$

Consequently we get $abcd=\sqrt{\tan \alpha \tan \beta \tan \gamma \tan \delta}\ge 3$