KöMaL 2011 B4370

Let $a$, $b$, $c$ denote the lengths of the sides of a triangle, and let $u$, $v$, $w$, respectively, be the distances of the centre of the incircle from the vertices opposite to the sides. Prove that $(a+b+c) \left(\frac{1}{u}+\frac{1}{v}+\frac{1}{w}\right)\le 3\left(\frac{a}{u}+\frac{b}{v}+\frac{c}{w}\right)$. (Suggested by J. Mészáros, Jóka)