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$x$,$y$ and $z$ are positive real numbers.

We have to show that $$\frac{y}{xy+2y+1} + \frac{z}{yz+2z+1} + \frac{x}{xz+2x+1}\leq\frac{3}{4}.$$ I am the equivalent of a senior in high school (not a US citizen) and we didn't take a multi-variable calculus course. Is there a way this could be solved without it?

(It's a problem I found in a math olympiad somewhere,but I can't remember which one.)

amWhy
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med chs
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1 Answers1

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The full expanding gives: $$\sum_{cyc}(x^2y^2z^2+2x^2y^2z-x^2yz+2x^2y-6xyz-xy+2x+1)\geq0,$$ which is true by AM-GM only: $$\sum_{cyc}(x^2y^2z^2+1-2xyz)\geq0,$$ $$\sum_{cyc}(x^2y-xyz)\geq0,$$ $$1.5\sum_{cyc}(x^2y^2z+z-2xyz)\geq0,$$ $$\frac{1}{2}\sum_{cyc}(x^2z^2y+x^2y-2x^2yz)\geq0$$ and $$\frac{1}{2}\sum_{cyc}(x^2y+y-2xy)\geq0.$$

Michael Rozenberg
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