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
=
\dots
=
=
It can be shown that sin sin
sin
=
, so
=
=
=
, so the answer is