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Averaging regular tetrahedra

Source: HMIC 2021 Problem 4

May 4, 2021
HMIC

Problem Statement

Let A1A2A3A4A_1A_2A_3A_4, B1B2B3B4B_1B_2B_3B_4, and C1C2C3C4C_1C_2C_3C_4 be three regular tetrahedra in 33-dimensional space, no two of which are congruent. Suppose that, for each i∈{1,2,3,4}i\in \{1,2,3,4\}, CiC_i is the midpoint of the line segment AiBiA_iB_i. Determine whether the four lines A1B1A_1B_1, A2B2A_2B_2, A3B3A_3B_3, and A4B4A_4B_4 must concur.
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