ICTM State 2015 Question 19

Geometry Level 4

In the diagraam shown, \(B\) lies on \(\overline{AC}\), \(AB=12, BC=12, \angle ADB = 15^\circ\) and \(\angle BDC = 30^\circ \).

If the length of \(AD\) can be expressed as \(k + w \sqrt f\), where \(k,w\) and \(f\) are positive integers with \(f\) square-free, find \(k+w+f\).

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