Problem:
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 vertices of the triangle partition the circle into three arcs with degree measures , , and . The chord fails to intersect the triangle if and only if both of the chosen points are within the same arc. This occurs with probability
Substituting transforms the equation into
The solutions for are and , and the corresponding solutions for are and , respectively, so the requested difference is .
The problems on this page are the property of the MAA's American Mathematics Competitions