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Found problems: 2

2021 XVII International Zhautykov Olympiad, #4
Tags: IZHO 2021 , izho , geometric inequality , geometry , inequalities
Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

2021 International Zhautykov Olympiad, 4
Tags: IZHO 2021 , izho , geometric inequality , geometry , inequalities
Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$