Difference between revisions of "2017 AIME I Problems/Problem 15"
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The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths <math>2\sqrt{3},~5,</math> and <math>\sqrt{37},</math> as shown, is <math>\frac{m\sqrt{p}}{n},</math> where <math>m,~n,</math> and <math>p</math> are positive integers, <math>m</math> and <math>n</math> are relatively prime, and <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p.</math> | The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths <math>2\sqrt{3},~5,</math> and <math>\sqrt{37},</math> as shown, is <math>\frac{m\sqrt{p}}{n},</math> where <math>m,~n,</math> and <math>p</math> are positive integers, <math>m</math> and <math>n</math> are relatively prime, and <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p.</math> | ||
| − | [asy] | + | <math>[asy] |
size(5cm); | size(5cm); | ||
pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); | pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); | ||
| Line 12: | Line 12: | ||
draw(A--B--C--A^^P--Q--R--P); | draw(A--B--C--A^^P--Q--R--P); | ||
dot(A--B--C--P--Q--R); | dot(A--B--C--P--Q--R); | ||
| − | [/asy] | + | [/asy]</math> |
==Solution== | ==Solution== | ||
==See Also== | ==See Also== | ||
Revision as of 17:56, 8 March 2017
Problem 15
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths 
and 
as shown, is 
where 
and 
are positive integers, 
and 
are relatively prime, and is not divisible by the square of any prime. Find 
$[asy] size(5cm); pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); real t = .385, s = 3.5*t-1; pair R = A*t+B*(1-t), P=B*s; pair Q = dir(-60) * (R-P) + P; fill(P--Q--R--cycle,gray); draw(A--B--C--A^^P--Q--R--P); dot(A--B--C--P--Q--R); [/asy]$ (Error compiling LaTeX. Unknown error_msg)
