Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.1 AIME, harder Find the answer

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 $\lfloor r \rfloor$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We are given $m \angle EOC = m \angle COD$ and $m \angle AOC + m \angle COD = 2m \angle EOC + m \angle COD = 90^{\circ}$ 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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