Perpendicular bisectors in a circle

Geometry Level 5

Let \(S\) be a set of \( 31 \) equally spaced points on a circle centered at \( O \), and consider a uniformly random pair of distinct points \( A \) and \( B \) (\(A, B \in S\)). The probability that the perpendicular bisectors of \( OA \) and \( OB \) intersect strictly inside the circle can be expressed as \( \frac{m}{n} \), where \( m,n \) are relatively prime positive integers. Find \( m+n \).

This problem is posed by Muhammad A.

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