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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