Difference between revisions of "2019 AIME II Problems/Problem 15"
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In acute triangle <math>ABC</math> points <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>C</math> to <math>\overline{AB}</math> and from <math>B</math> to <math>\overline{AC}</math>, respectively. Line <math>PQ</math> intersects the circumcircle of <math>\triangle ABC</math> in two distinct points, <math>X</math> and <math>Y</math>. Suppose <math>XP=10</math>, <math>PQ=25</math>, and <math>QY=15</math>. The value of <math>AB\cdot AC</math> can be written in the form <math>m\sqrt n</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. | In acute triangle <math>ABC</math> points <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>C</math> to <math>\overline{AB}</math> and from <math>B</math> to <math>\overline{AC}</math>, respectively. Line <math>PQ</math> intersects the circumcircle of <math>\triangle ABC</math> in two distinct points, <math>X</math> and <math>Y</math>. Suppose <math>XP=10</math>, <math>PQ=25</math>, and <math>QY=15</math>. The value of <math>AB\cdot AC</math> can be written in the form <math>m\sqrt n</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. | ||
− | ==Solution== | + | ==Solution 1== |
Let <math>AP=a, AQ=b, \cos\angle A = k</math> | Let <math>AP=a, AQ=b, \cos\angle A = k</math> | ||
Revision as of 02:30, 16 February 2021
Contents
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 integers, and
is not divisible by the square of any prime. Find
.
Solution 1
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
By SpecialBeing2017
Solution 2
Let and
By power of point, we have
and
Therefore, substituting in the values:
Notice that quadrilateral is cyclic.
From this fact, we can deduce that and
Therefore is similar to
.
Therefore:
Now using Law of Cosines on we get:
Notice
Substituting and Simplifying:
Now we solve for using regular algebra which actually turns out to be very easy.
We get and from the above relations between the variables we quickly determine
,
and
Therefore
So the answer is
By asr41
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.