Fun of Thinking — Math Problems from Around the World

In a regular $n$ -gonal pyramid $P\text{-}A_1A_2\cdots A_n$ (where $n \geqslant 3$ ), let $O$ be the center of the regular $n$ -gon base $A_1A_2\cdots A_n$ , and let $B$ be the midpoint of edge $A_1A_n$ .

1. Prove that

PO^2 \sin^2\frac{\pi}{n} + PA_1^2 \cos^2\frac{\pi}{n} = PB^2.

Let $\alpha$ be the angle between a lateral edge and the base, and $\beta$ the angle between a lateral face and the base. Determine which of

\frac{1}{n}\sum_{i=1}^{n} \cos\angle A_iPB \qquad \text{and} \qquad \sin\alpha\,\sin\beta

is larger, and prove your claim.

Difficulty	7/10
Topics	trigonometry, solid geometry, Symmetry, vectors

A Pyramid Invariant

Problem

Answer

Solution

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