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# Nested Figures in a Pentagon

## Question

Joining every vertex of a convex pentagon to every other vertex forms three nested figures:
the original outer pentagon
an inner pentagon
a five-pointed star between them
Suppose the perimeters of these three figures are all primes; show that the three primes’ sum is at least 20.

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no clue about this one. seems to me the lowest prime number the outside could be is 11, with the blue diamond being 7 and the black pentagon being 3.

but how to prove that is beyond me.

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no clue about this one. seems to me the lowest prime number the outside could be is 11, with the blue diamond being 7 and the black pentagon being 3.

but how to prove that is beyond me.

When you said the blue diamond, did you mean the blue star? If so, it would have a larger perimeter than the red pentagon. But that is all I can determine.

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