MathDB

Let

A,B,C

be angles of an acute triangle with \begin{align*} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}. \end{align*} 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

.
Note: due to an oversight by the exam-setters, there is no acute triangle satisfying these conditions. You should instead assume

ABC

is obtuse with

\angle B > 90^{\circ}

Problem Statement