2023 WSMO Accuracy Round Problems/Problem 10

Problem

In tetrahedron $T$ of side length $12,$ let $S_1$ be the sphere inscribed in $T$ and let $S_2$ be the sphere circumscribed around $T.$ Let $R$ be a rectangular prism such that all points on $S_1$ lie strictly inside or are touching $R$ and all points on $R$ lie strictly inside or are touching $S_2.$ The minimum possible volume of $R$ is $m\sqrt{n}.$ Find $m+n.$

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2023 WSMO Accuracy Round Problems/Problem 10

Problem

Solution