3. [8] A semicircle with radius 2021 has diameter $A B$ and center $O$. Points $C$ and $D$ lie on the semicircle such that $\angle A O C<\angle A O D=90^{\circ}$. A circle of radius $r$ is inscribed in the sector bounded by $O A$ and $O C$ and is tangent to the semicircle at $E$. If $C D=C E$, compute $\lfloor r\rfloor$.
Answer: 673

Solution: We are given
$$
m \angle E O C=m \angle C O D
$$
and
$$
m \angle A O C+m \angle C O D=2 m \angle E O C+m \angle C O D=90^{\circ} .
$$

So $m \angle E O C=30^{\circ}$ and $m \angle A O C=60^{\circ}$. Letting the radius of the semicircle be $R$, we have
$$
(R-r) \sin \angle A O C=r \Rightarrow r=\frac{1}{3} R,
$$
so
$$
\lfloor r\rfloor=\left\lfloor\frac{2021}{3}\right\rfloor=673 \text {. }
$$