2020 AIME I Problems/Problem 13
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Problem
Point lies on side
of
so that
bisects
The perpendicular bisector of
intersects the bisectors of
and
in points
and
respectively. Given that
and
the area of
can be written as
where
and
are relatively prime positive integers, and
is a positive integer not divisible by the square of any prime. Find
Please thank awang11 for his amazing diagram:
Solution
Points are defined as shown. It is pretty easy to show that by spiral similarity at
by some short angle chasing. Now, note that
is the altitude of
, as the altitude of
. We need to compare these altitudes in order to compare their areas. Note that Stewart's theorem implies that
, the altitude of
. Similarly, the altitude of
is the altitude of
, or
. However, it's not too hard to see that
, and therefore
. From here, we get that the area of
is
, by similarity. ~awang11
See Also
2020 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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