2017 AIME I Problems/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 .
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 This simplifies to Which factors as So . The difference between these is .
We actually do not need to spend time factoring . Since the problem asks for , where and are the roots of the quadratic, we can utilize Vieta's by noting that . Vieta's gives us and Plugging this into the above equation and simplifying gives us or .
Our answer is then .
https://youtu.be/Mk-MCeVjSGc?t=690 ~Shreyas S
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