Question 108
Let $ABC$ be a triangle. The points $K$, $L$ and $M$ lie on the sides $BC$, $CA$ and $AB$ respectively, such that the lines $AK$, $BL$ and $CM$ meet in one point. Show that one can choose two of the triangles $AML$, $BKM$ and $CLK$ such that the sum of their inradii is at least as large as the inradius of triangle $ABC$.