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2

Part of 1955 Czech and Slovak Olympiad III A

Problems(1)

Cuboid between spheres

Source: Czech and Slovak Olympiad 1955, National Round, Problem 2

8/15/2022
Let S1,S2\mathsf{S}_1,\mathsf{S}_2 be concentric spheres with radii a,ba,b respectively, where a<b.a<b. Denote ABCDA′B′C′D′ABCDA'B'C'D' a square cuboid (ABCD,A′B′C′DABCD,A'B'C'D are the squares and AA′∥BB′∥CC′∥DD′AA'\parallel BB'\parallel CC'\parallel DD') such that A,B,C,D∈S2A,B,C,D\in\mathsf{S}_2 and the plane A′B′C′D′A'B'C'D' is tangent to S1.\mathsf{S}_1. Finally assume that ABAA′=ab.\frac{AB}{AA'}=\frac ab. Compute the lengths AB,AA′.AB,AA'. How many of such cuboids exist (up to a congruence)?
geometry3D geometrysphereprism