{"status": "success", "data": {"description_md": "Let $A,B,C$ be angles of an acute triangle with
\n$$\\cos^2 A + \\cos^2 B + 2 \\sin A \\sin B \\cos C = \\frac{15}{8}$$ and\n$$\\cos^2 B + \\cos^2 C + 2 \\sin B \\sin C \\cos A = \\frac{14}{9}.$$
\nThere 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 $\\angle B > 90^{\\circ}$.\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

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.

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}.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "2013 AIME II Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/13_aime_II_p14"}}