1985 Putnam B4

Let $C$ be the unit circle $x^2+y^2=1$. A point $p$ is chosen randomly

on the circumference $C$ and another point $q$ is chosen randomly from

the interior of $C$ (these points are chosen independently and

uniformly over their domains). Let $R$ be the rectangle with sides

parallel to the $x$ and $y$-axes with diagonal $pq$. What is the

probability that no point of $R$ lies outside of $C$?

设$C$为单位圆$x^2+y^2=1$。随机选择一个点 $p$

在圆周 $C$ 上，另一个点 $q$ 是随机选择的

$C$ 的内部(这些点是独立选择的并且

在其域上统一)。设 $R$ 为有边的矩形

与 $x$ 和 $y$ 轴平行，对角线 $pq$。什么是

$R$ 中没有点位于 $C$ 之外的概率？