Stars Within Stars

Geometry Level 4

Let \(\{p/q\}\) denote the \(p\)-pointed star formed by joining every \(q^{\text{th}}\) vertex on a convex \(p\)-gon until you reach the starting point. This is repeated with different starting points if necessary until we form a \(p\)-pointed star.

Is it true that \(\{p/q\}\) contains \(\{p/r\}\) if \(r \le q\)?

Note:

  • \(\{p/q\}\) is defined for all \(p \ge 3\), and \(1 \le q < \frac p2\).

  • As an explicit example, the following shows three possible \(7\)-pointed stars. Here \(\{7/2\}\) contains \(\{7/1\}\) because there is an instance of \(\{7/1\}\) inside \(\{7/2\}\).

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