Virginia Mathematics Teacher Fall 2016

So l ut i ons t o Spr i ng 2016 HEXA Cha l l enge Probl ems September Challenge:

Inside a sphere (the original print had a typo that said circle) there is a polyhedron where all n vertices are on the sphere. Prove that the number of points where the diagonals intersect each other, cannot exceed:

SOLUTION : The number of quadrilaterals that can be formed out of n points on the circle is:

Each of the quadrilaterals has two diagonals, therefore they can intersect at only one point. As a result, the total number of points of intersection for the polyhedron can never exceed this number.

Virginia Mathematics Teacher vol. 43, no. 1

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