2010 AIME I Problems/Problem 15
Problem
In with
,
, and
, let
be a point on
such that the incircles of
and
have equal radii. Let
and
be positive relatively prime integers such that
. Find
.
Solution
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Let and
so
. Let the incenters of
and
be
and
respectively, and their equal inradii be
. From
, we find that
Let the incircle of meet
at
and the incircle of
meet
at
. Then note that
is a rectangle. Also,
is right because
and
are the angle bisectors of
and
respectively and
. By properties of tangents to circles
and
. Now notice that the altitude of
to
is of length
, so by similar triangles we find that
(3). Equating (3) with (1) and (2) separately yields
and adding these we have
See also
2010 AIME I (Problems • Answer Key • Resources) | ||
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