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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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Are a,b,c,d $> 0$? – Gautam Shenoy Nov 14 '12 at 16:33
I don't see the linear algebra here. – Julian Kuelshammer Nov 14 '12 at 16:33
it doesn't say that a,b,c,d >0, but let's pressume they are. can you solve it if they are? – Dekac Nov 14 '12 at 16:46

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

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how about if we added a condition a,b,c,d>1 ? Would this be solvable? – Dekac Nov 14 '12 at 17:41
@Dekac: The inequality is scale invariant, so it is also false for $a=b=4, c=d=2$. – robjohn Nov 14 '12 at 18:03

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