WMC 2018 Pursuit - Pack 2, B2

Let $\angle BAD=\theta$ . Then $\angle BAC=2\theta$ .

By the law of cosines, $8^2=3^2+6^2-2\cdot 3\cdot 6\cos{\angle BAC}$ . This can be simplified to $\cos{\angle BAC}=-\frac{19}{36}\implies \cos{2\theta}=-\frac{19}{36}$ .

Therefore, $\cos{2\theta}=2\cos^2{\theta}-1=-\frac{19}{36}\implies \cos^2{\theta}=\frac{17}{72}$

Finally, $\cos{\theta}=\frac{\sqrt{17}}{6\sqrt{2}}=\frac{\sqrt{34}}{12}\implies a+b=46$ .