2017 AIME I Problems/Problem 6
Problem 6
A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure 
. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is 
. Find the difference between the largest and smallest possible values of .
Solution
The probability that the chord doesn't intersect the triangle is 
. The only way this can happen is if the two points are chosen on the same arc between two of the triangle vertices. The probability that a point is chosen on one of the arcs opposite one of the base angles is 
, and the probability that a point is chosen on the arc between the two base angles is 
. Therefore, we can write ![\[2\left(\frac{x}{180}\right)^2+\left(\frac{180-2x}{180}\right)^2=\frac{11}{25}\]](http://latex.artofproblemsolving.com/9/4/a/94a34891cd57d740d5d4855be13d12264108e949.png)
This simplifies to ![\[x^2-120x+3024=0\]](http://latex.artofproblemsolving.com/7/c/e/7cef2414362e3fd2f1690e612a88a47f3d8ed6aa.png)
Which factors as ![\[(x-84)(x-36)=0\]](http://latex.artofproblemsolving.com/b/8/2/b8235fc1398aca96e8022908afcaea96a6f0b1d4.png)
So 
. The difference between these is 
.
See Also
| 2017 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 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. 
