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Omni-MATH / In three-dimensional space, let S be the region of points (x,y,z) satisfying −1≤z≤1. Let S₁,S₂,…,S₂₀₂₂ be 2022 independent random rotations of S about the origin ( 0,0,0). The expe…

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problem

In three-dimensional space, let SS be the region of points (x,y,z)(x, y, z) satisfying −1≤z≤1-1 \leq z \leq 1. Let S1,S2,…,S2022S_{1}, S_{2}, \ldots, S_{2022} be 2022 independent random rotations of SS about the origin ( 0,0,00,0,0). The expected volume of the region S1∩S2∩⋯∩S2022S_{1} \cap S_{2} \cap \cdots \cap S_{2022} can be expressed as aπb\frac{a \pi}{b}, for relatively prime positive integers aa and bb. Compute 100a+b100 a+b.
Plain-text mathematical notation (without MathML)
In three-dimensional space, let S be the region of points (x,y,z) satisfying −1≤z≤1. Let S₁,S₂,…,S₂₀₂₂ be 2022 independent random rotations of S about the origin ( 0,0,0). The expected volume of the region S₁∩S₂∩⋯∩S₂₀₂₂ can be expressed as (aπ)/(b), for relatively prime positive integers a and b. Compute 100a+b.
Original LaTeX notation
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 $100 a+b$.

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task_id
bb3c3aa8-0a92-5075-8c89-25ab77fa7da2
task_key
test--bb3c3aa8-0a92-5075-8c89-25ab77fa7da2
task_revision_id
1
config / split
/ test

Official source

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