Problem:
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
Answer Choices:
A.
B.
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D.
E.
Solution:
Three chords create a triangle if and only if they intersect pairwise inside the circle. Two chords intersect inside the circle if and only if their endpoints alternate in order around the circle. Therefore, if points , and are in order around the circle, then only the chords , all intersect pairwise inside the circle. Thus every set of points determines a unique triangle, and there are such triangles.
The problems on this page are the property of the MAA's American Mathematics Competitions