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$\frac{1}{\frac{1}{a}+\frac{1}{b}}+\frac{1}{\frac{1}{c}+\frac{1}{d}} \leq \frac{1}{\frac{1}{a+b}+\frac{1}{c+d}}$

I think it has something to do with Harmonic mean, but can't fighre it out.

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    it doesn't say that a,b,c,d >0, but let's pressume they are. can you solve it if they are?2012-11-14

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This is wrong. For example, take $a=b=2$, $c=d=1$

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    @Dekac: The inequality is scale invariant, so it is also false for $a=b=4, c=d=2$.2012-11-14
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It should be $\frac{1}{\frac{1}{a}+\frac{1}{b}}+\frac{1}{\frac{1}{c}+\frac{1}{d}} \leq \frac{1}{\frac{1}{a+d}+\frac{1}{b+c}}$ for positives $a$, $b$ $c$ and $d$, which is $(ac-bd)^2\geq0$.