# An Inequality in Triangle with Roots and Circumradius

### Problem

### Solution

Use *Hölder's inequality* followed by the rearrangement inequality:
,

$\displaystyle\begin{align} (a\sqrt{b}+b\sqrt{c}+c\sqrt{a})^2 &\le (a+b+c)(ab+bc+ca)\\ &\le 2s(a^2+b^2+c^2). \end{align}$

But we know that $a^2+b^2+c^2\le 9R^2.\,$ A combination of the two gives the desired result.

### Acknowledgment

Dan Sitaru has kindly posted the above problem from his book *Math Accent*, with a solution, at the CutTheKnotMath facebook page. The solution is by Mihalcea Andrei Stefan, a grade 9 student.

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