Question 64
Points $K$, $L$, $M$ lie on the sides $BC$, $CA$, $AB$ of a triangle $ABC$, respectively, so that $AK$, $BL$, $CM$ meet at a common point. Let $r_1$, $r_2$, $r_3$, $r$ be the inradii of the triangles $ALM$, $BMK$, $CKL$, $ABC$, respectively.
Prove that $r_1 + r_2 + r_3 \ge r$ or $r_2 + r_3 \ge r$ or $r_1 \ge r$.
Editorial metadata note: Metadata discrepancy: the dataset year is 2014, while booklet_year is 2015. The title follows dataset classification, not a verified Belarus 2014 event date.
Prove that $r_1 + r_2 + r_3 \ge r$ or $r_2 + r_3 \ge r$ or $r_1 \ge r$.
Editorial metadata note: Metadata discrepancy: the dataset year is 2014, while booklet_year is 2015. The title follows dataset classification, not a verified Belarus 2014 event date.