2014 IMO Shortlist G2

Let $A B C$ be a triangle. The points $K, L$, and $M$ lie on the segments $B C, C A$, and $A B$, respectively, such that the lines $A K, B L$, and $C M$ intersect in a common point. Prove that it is possible to choose two of the triangles $A L M, B M K$, and $C K L$ whose inradii sum up to at least the inradius of the triangle $A B C$. (Estonia)

设 $A B C$ 为三角形。点$K、L$和$M$分别位于线段$B C、C A$和$A B$上，使得线$A K、B L$和$C M$相交于公共点。证明可以选择三角形 $A L M、B M K$ 和 $C K L$ 中的两个，其内半径之和至少等于三角形 $A B C$ 的内半径。 (爱沙尼亚)