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well known but yet very hard solid geometry problem

Source: Romanian Nationals RMO 2005 - grade 10, problem 2, [Arthur Engel, Problem-Solving St., problem 3.30]

March 31, 2005
geometry3D geometrypyramidtrigonometrygeometric transformationrotationconics

Problem Statement

The base A1A2…AnA_{1}A_{2}\ldots A_{n} of the pyramid VA1A2…AnVA_{1}A_{2}\ldots A_{n} is a regular polygon. Prove that if ∠VA1A2≡∠VA2A3≡⋯≡∠VAn−1An≡∠VAnA1,\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1}, then the pyramid is regular.
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