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HMMT Spring Guts Round, 2021

Geometry Difficulty 5.1 AIME, harder Prove it United States

Problem:

A semicircle with radius $2021$ has diameter $AB$ and center $O$ . Points $C$ and $D$ lie on the semicircle such that $\angle AOC < \angle AOD = 90^\circ$ . A circle of radius $r$ is inscribed in the sector bounded by $OA$ and $OC$ and is tangent to the semicircle at $E$ . If $CD = CE$ , compute $\lfloor r \rfloor$ .

Solution

Solution:

We are given
$m \angle EOC = m \angle COD$
and
$m \angle AOC + m \angle COD = 2 m \angle EOC + m \angle COD = 90^\circ.$
So $m \angle EOC = 30^\circ$ and $m \angle AOC = 60^\circ$ . Letting the radius of the semicircle be $R$ , we have
$(R - r) \sin \angle AOC = r \Rightarrow r = \frac{1}{3} R$
so
$\lfloor r \rfloor = \left\lfloor \frac{2021}{3} \right\rfloor = 673$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.