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UKMT Pink Kangaroo, 2022

Geometry Difficulty 3.0 AMC 10/12 Find the answer United Kingdom

The cube shown has sides of length $2$ units. Holes in the shape of a hemisphere are carved into each face of the cube. The six hemispheres are identical and their centres are at the centres of the faces of the cube. The holes are just large enough to touch the hole on each neighbouring face. What is the diameter of each hole?

(A) $1$ (B) $\sqrt{2}$ (C) $2-\sqrt{2}$ (D) $3-\sqrt{2}$ (E) $3-\sqrt{3}$

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