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
This simplifies to
Which factors as
So
. The difference between these is
.