HMMT February

Geometry Difficulty 4.8 AIME Find the answer United States

Problem:

In three-dimensional space, let $S$ be the region of points $(x, y, z)$ satisfying $-1 \leq z \leq 1$ . Let $S_{1}, S_{2}, \ldots, S_{2022}$ be 2022 independent random rotations of $S$ about the origin $(0,0,0)$ . The expected volume of the region $S_{1} \cap S_{2} \cap \cdots \cap S_{2022}$ can be expressed as $\frac{a \pi}{b}$ , for relatively prime positive integers $a$ and $b$ . Compute $100a + b$ .

Your answer

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

Solution

Solution:

Consider a point $P$ of distance $r$ from the origin. The distance from the origin of a random projection of $P$ onto a line is uniform from $0$ to $r$ . Therefore, if $r < 1$ then the probability of $P$ being in all the sets is $1$ , while for $r \geq 1$ it is $r^{-2022}$ . Therefore the volume is
$\frac{4 \pi}{3} + 4 \pi \int_{1}^{\infty} r^{2} r^{-2022} d r = 4 \pi \left(\frac{1}{3} + \frac{1}{2019}\right) = \frac{2696 \pi}{2019}$

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