IMO 1979 LL BUL11

Prove that a pyramid A1A2 . . . A2k+1S with equal lateral edges

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IMO 1979 LL BUL11

Origin: BUL

Problem

Prove that a pyramid A1A2 . . . A2k+1S with equal lateral edges and equal space angles between adjacent lateral walls is regular. Variant. Prove that a pyramid A1 . . . A2k+1S with equal space angles between adjacent lateral walls is regular if there exists a sphere tangent to all its edges.