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Exercise 1.4 from a great book The Cauchy-Schwarz Master Class asks to prove the following:

For all positive $x$, $y$ and $z$, one has $$x+y+z \leq 2 \left(\frac{x^2}{y+z} + \frac{y^2}{x+z} + \frac{z^2}{x+y}\right).$$

Introduction to the exercise says:

There are many situations where Cauchy's inequality conspires with symmetry to provide results that are visually stunning.

How to prove that inequality? And how does one benefit from the "symmetry"? What is the general idea behind this "conspiracy"?

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    The back of the book has solutions to the exercises.2011-10-10
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    (This was my original comment, that I removed by mistake.) I don't think the answers here will explain the conspiracy better than Steele does in his book. (But let's hope that I am wrong about this.) So my recommendation is: *read the book*! :)2011-10-10
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    Uh, I didn't even realize that the book has solutions. I've only printed out the first chapter, but the book is definitely worth buying. Unfortunately, as Martin has noticed, the back of the book doesn't elaborate on the conspiracy.2011-10-10

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