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Weekly Problem 15 - 2012

Stage: 3 Short Challenge Level: Challenge Level:1

It can possess all five properties.
The description does not say that heptagon has to be convex, i.e. all of its interior angles need not be less than $180^{\circ}$. Since $(2 \timesĀ 7 - 4) \times 90 = 900$, the interior angles of all heptagons total $900^{\circ}$. The creation of a heptagon with all the given conditions is possible as the diagram shows. Notice that four of the interior angles are acute and the other three are reflex angles.

This problem is taken from the UKMT Mathematical Challenges.

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