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Geometry Difficulty 8.3 Shortlist Prove it Hong Kong

A circle is circumscribed around an isosceles triangle whose two base angles are equal to x∘x^\circ. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is 1425\frac{14}{25}. Find the sum of the largest and smallest possible value of xx.

Solution

The answer is 120120.

The probability that the chord does not intersect the triangle is 1125\frac{11}{25}. The only way this can happen is when 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 to one of the base angles is x180\frac{x}{180} and the probability that a point is chosen on the arc between the two base angles is 180−2x180\frac{180 - 2x}{180}. Therefore we have
2(x180)2+(180−2x180)2=1125. 2\left(\frac{x}{180}\right)^2 + \left(\frac{180 - 2x}{180}\right)^2 = \frac{11}{25}.
This simplifies to x2−120x+3024=0x^2 - 120x + 3024 = 0, and can be factorized as (x−84)(x−36)=0(x-84)(x-36) = 0. Hence the sum of roots is 84+36=12084 + 36 = 120.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.