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== | ||
+ | First we have <math>a\cos A=PQ=25</math>, and <math>(a\cos A)(c\cos C)=(a\cos C)(c\cos A)=AP\cdot PB=10(25+15)=400.</math> Similarly, <math>(a\cos A)(b\cos B)=15(10+25)=525,</math> and dividing these each by <math>a\cos A</math> gives | ||
+ | <math>b\cos B=21,c\cos C=16</math>. | ||
+ | |||
+ | |||
+ | It is known that the sides of the orthic triangle are <math>a\cos A,b\cos B,c\cos C</math>, and its angles are <math>\pi-2A</math>,<math>\pi-2B</math>, and <math>\pi-2C</math>. We thus have the three sides of the orthic triangle now. | ||
+ | Letting <math>D</math> be the foot of the altitude from <math>A</math>, we have, in <math>\triangle DPQ</math>, | ||
+ | <cmath>\cos P,\cos Q=\frac{21^2+25^2-16^2}{2\cdot 21\cdot 25},\frac{16^2+25^2-21^2}{2\cdot 16\cdot 25}=27/35,11/20.</cmath> | ||
+ | <cmath>\Rightarrow \cos B=\cos\biggl(\frac{\pi-P}{2}\biggr)=\sin\frac{P}{2}=\sqrt{4/35},</cmath> | ||
+ | similarly, we get | ||
+ | <cmath>\cos C=\cos\biggl(\frac{\pi-Q}{2}\biggr)=\sin\frac{Q}{2}=\sqrt{9/40}.</cmath> | ||
+ | Our final answer is then <cmath>bc= \frac{(b\cos B)(c\cos C)}{\cos B\cos C}=\frac{16\cdot 21}{6/sqrt{1400}}</cmath> | ||
+ | <cmath>=56\cdot\sqrt{1400}=560\sqrt{14}.</cmath> | ||
+ | |||
+ | The requested sum is <math>\boxed{574}</math>. | ||
+ | |||
+ | ༺\\crazyeyemoody9❂7//༻ | ||
==Solution 1== | ==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 17:33, 3 March 2021
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
First we have , and Similarly, and dividing these each by gives .
It is known that the sides of the orthic triangle are , and its angles are ,, and . We thus have the three sides of the orthic triangle now.
Letting be the foot of the altitude from , we have, in ,
similarly, we get
Our final answer is then
The requested sum is .
༺\\crazyeyemoody9❂7//༻
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 |
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