Question

A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x.$ 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 $\frac{14}{25}.$ Find the difference between the largest and smallest possible values of $x.$

Answer

Solution:

Each inscribed angle of the triangle subtends an arc of twice its measure, so the vertices split the circle into arcs of $2x,$ $2x,$ and $360 - 4x$ degrees. The chord fails to intersect the triangle exactly when both random points fall in the same arc, which has probability $\left(\frac{2x}{360}\right)^2 + \left(\frac{2x}{360}\right)^2 + \left(\frac{360 - 4x}{360}\right)^2 = 1 - \frac{14}{25} = \frac{11}{25}.$

Setting $y = \frac{x}{180},$ this reads $2y^2 + (1 - 2y)^2 = \frac{11}{25},$ which simplifies to $75y^2 - 50y + 7 = 0,$ with roots $y = \frac{1}{5}$ and $y = \frac{7}{15}.$ These give $x = 36$ and $x = 84,$ both legitimate base angles of an isosceles triangle.

The requested difference is $84 - 36 = 48.$

2017 AIME I Problem 6

Solution:

Problem 6 in Other Years