2009 AIME II Problems/Problem 13
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Problem
Let and be the endpoints of a semicircular arc of radius . The arc is divided into seven congruent arcs by six equally spaced points , , , . All chords of the form or are drawn. Let be the product of the lengths of these twelve chords. Find the remainder when is divided by .
Solution
Let be the midpoint of and . Assume is closer to instead of . = . Using the Law of Cosines,
= = . . . =
So = . It can be rearranged to form
= .
= - , so we have
=
=
=
It can be shown that sin sin sin = , so = = = , so the answer is