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Prove that $\left (\dfrac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$ if $a$, $b$ and $c$ are distinct natural numbers. Is it possible using induction?

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2 Answers 2

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I recommend to read the answer from here. section Weighted AM-GM Inequality :) it is a good one .

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  • $\begingroup$ nice link! (+1) $\endgroup$ Aug 23, 2012 at 19:14
  • $\begingroup$ @downvoter why (-1)? $\endgroup$
    – Iuli
    Feb 5, 2013 at 12:56
  • $\begingroup$ wasn't me but I guess because the linked page is no longer available $\endgroup$
    – Valentin
    Feb 10, 2013 at 0:46
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    $\begingroup$ @Iuli Answers that only include a link are bad, for exactly this reason - very often the page that was linked to disappears. Can you edit your answer so that it's useful again? $\endgroup$ Feb 13, 2013 at 12:19
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Rewrite it as: $$ \left( a \frac{a}{a+b+c} + b \frac{b}{a+b+c} + c \frac{c}{a+b+c} \right) > a^\frac{a}{a+b+c} \cdot b^\frac{b}{a+b+c} \cdot c^\frac{c}{a+b+c} $$ This is Jensen's inequality: $$ \log\left(\mathsf{E}\left(X\right)\right) > \mathsf{E}\left(\log\left(X\right)\right) \quad \text{or}\quad \mathsf{E}\left(X\right) > \exp \left( \mathsf{E}\left(\log\left(X\right)\right) \right) $$ where $X$ is the random variable which can assume one of three possible values $\{a,b,c\}$ with respective probabilities $\{ \frac{a}{a+b+c}, \frac{b}{a+b+c}, \frac{c}{a+b+c} \}$.

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  • $\begingroup$ @Fixee Thanks for the edits. $\endgroup$
    – Sasha
    Aug 23, 2012 at 16:45
  • $\begingroup$ Nice solution! (+1) $\endgroup$ Aug 23, 2012 at 19:15

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