Proving inequality with squares – math.stackexchange.com

Proving inequality with squares – math.stackexchange.com

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I need to prove that for all $x,y\geq1$ : $$\frac{x+y+1}{\sqrt{x^2+x}+\sqrt{y^2+y}}\leq2$$ I have tried assuming that $$\frac{x+y+1}{\sqrt{x^2+x}+\sqrt{y^2+y}}>2$$, and get some contradiction, ...

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