5 Days to 2016 Make it and Bake it

Geometry Level 4

If in a triangle \(ABC\) an angle bisector is drawn from \(A\) to \(BC\) such that it cuts \(BC\) at \(D\). If \( \angle BAD = 15^{\circ} \) and \( \angle ABC =45^{o} \). If \(AD=5\sqrt{6}\) and the area of the triangle can be expressed as \( \large{ \frac{m}{\sqrt{n}+p} } \), for some positive integers \( (m, n, p) \). Find \(m+n+p\).

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