Mock AIME 1 2010 Problems/Problem 13
Problem
Suppose is inscribed in circle
.
and
are the feet of the altitude from
to
and
to
, respectively. Let
be the intersection of lines
and
, let
be the point of intersection of
and line
distinct from
, and let
be the foot of the perpendicular from
to
. Given that
,
, and
, and that
can be expressed in the form
, where
and
are relatively prime positive integers and
is an integer not divisible by the square of any prime, find the last three digits of
.
Solution
Let . Because the problem gives us
, we think to use the Law of Cosines in
, which yields
. Subtituting the values given by the problem, we get
, which gives
.
To find another expression for , we think of the formula
. We know that the area of the triangle is
. Substituting this in the previous equation for
, we get that
, so
.
Setting these two expressions for equal to each other reveals that
, so
by the identity
is supplementary to
, and
is supplementary to
, because
is a cyclic quadrilateral. Thus,
, so
. Thus,
, so our answer is
.
See Also
Mock AIME 1 2010 (Problems, Source) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |