TrigInTriangle AIME Difficult
2013

Problem - 209
Let $A,B,C$ be angles of an acute triangle with $$\cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C = \frac{15}{8}$$ and $$\cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A = \frac{14}{9}$$ There are positive integers $p$, $q$, $r$, and $s$ for which $\cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s},$ where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p+q+r+s$.

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