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JamesJ
 4 years ago
Who's up for a challenge? Show that for all positive \( x, y, z \) we have
\[ \left(\frac{x+y}{x+y+z}\right)^{1/2} + \left(\frac{x+z}{x+y+z}\right)^{1/2} + \left(\frac{y+z}{x+y+z}\right)^{1/2} \leq \ 6^{1/2} \]
JamesJ
 4 years ago
Who's up for a challenge? Show that for all positive \( x, y, z \) we have \[ \left(\frac{x+y}{x+y+z}\right)^{1/2} + \left(\frac{x+z}{x+y+z}\right)^{1/2} + \left(\frac{y+z}{x+y+z}\right)^{1/2} \leq \ 6^{1/2} \]

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anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0I can reduce it down to \[(x+y)^{1/2}(y+z)^{1/2} + (x+y)^{1/2}(x + z)^{1/2} + (y+z)^{1/2}(x+z)^{1/2} \le 2(x+y+z)\]

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0x=2 y=2 z=2 just one example

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.5Hint: Use the CauchySchwarz Inequality

nikvist
 4 years ago
Best ResponseYou've already chosen the best response.1\[\frac{1}{2}\left(x+y+\frac{1}{x+y+z}\right)\ge\sqrt{\frac{x+y}{x+y+z}}\]\[\frac{1}{2}\left(y+z+\frac{1}{x+y+z}\right)\ge\sqrt{\frac{y+z}{x+y+z}}\]\[\frac{1}{2}\left(z+x+\frac{1}{x+y+z}\right)\ge\sqrt{\frac{z+x}{x+y+z}}\]\[\sum\quad\Rightarrow\quad\frac{1}{2}\left(2(x+y+z)+\frac{3}{x+y+z}\right)\ge\]\[\ge\sqrt{\frac{x+y}{x+y+z}}+\sqrt{\frac{y+z}{x+y+z}}+\sqrt{\frac{z+x}{x+y+z}}\]\[u=x+y+z\quad,\quad f(u)=u+\frac{3}{2u}\quad,\quad f'(u)=1\frac{3}{2u^2}=0\]\[u=\sqrt{\frac{3}{2}}\quad\Rightarrow\quad f_{\min}=\sqrt{6}\]

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.5Nicely done. I'll wait a bit and see what other proofs turn up, if any, and then I'll post my solution.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0I think I would have used AMGM as shown by nikvist.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.5I'm going to write this out. It's going to take a few minutes ...

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.5\[ \left( \frac{x+y}{x+y+z} \right)^{1/2} + \left( \frac{y+z}{x+y+z} \right)^{1/2} + \left( \frac{z+x}{x+y+z} \right)^{1/2} \] \[ = 1.\left( \frac{x+y}{x+y+z} \right)^{1/2} + 1.\left( \frac{y+z}{x+y+z} \right)^{1/2} + 1.\left( \frac{z+x}{x+y+z} \right)^{1/2} \] Now by CauchySchwatz this expression is \[ \leq (1^2 + 1^2 + 1^2)^{1/2} . \left( \frac{x+y}{x+y+z} + \frac{y+z}{x+y+z} + \frac{z+x}{x+y+z} \right)^{1/2} \] \[ = 3^{1/2} \left( \frac{2(x+y+z)}{x+y+z} \right)^{1/2} \] \[ = 6^{1/2} \] qed

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0That's even more compact, well done James.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0To ping somone use @ feature, for example @JamesJ

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0fool got your wing clipped
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