2019 AIME II Problems/Problem 15
Problem
In acute triangle points and are the feet of the perpendiculars from to and from to , respectively. Line intersects the circumcircle of in two distinct points, and . Suppose , , and . The value of can be written in the form where and are positive relatively prime integers. Find .
Solution
Let
Therefore
By power of point, we have Which are simplified to
Or
(1)
Or
Let Then,
In triangle , by law of cosine
Pluging (1)
Or
Substitute everything by
The quadratic term is cancelled out after simplified
Which gives
Plug back in,
Then
So the final answer is 560 + 14 = 574
By SpecialBeing2017
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
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