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Belarus — Selection and Training Session

Geometry Difficulty 5.9 AIME, harder Prove it Belarus

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 $r_1 \ge r$ .

Solution

2. See IMO-2014 Shortlist, Problem G2.

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