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3 circles inscribed in a semicirlce, radii inequalities (1995 Chile Level 2 P7)

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August 29, 2019
Geometric Inequalitiesradiuscirclesemicircle

Problem Statement

In a semicircle of radius 44 three circles are inscribed, as indicated in the figure. Larger circles have radii R1 R_1 and R2 R_2 , and the larger circle has radius r r . a) Prove that 1r=1R1+1R2 \dfrac {1} {\sqrt{r}} = \dfrac {1} {\sqrt{R_1}} + \dfrac {1} {\sqrt{R_2}} b) Prove that R1+R2≤8(2−1) R_1 + R_2 \le 8 (\sqrt{2} -1) c) Prove that r≤2−1 r \le \sqrt{2} -1 https://cdn.artofproblemsolving.com/attachments/0/9/aaaa65d1f4da4883973751e1363df804b9944c.jpg
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