2025 AIME II Problem 12

Let A_1A_2\ldots A_{11} be an 11-sided non-convex simple polygon with the following properties:

• For every integer 2 \leq i \leq 10, the area of A_iA_1A_{i+1} is 1,

• For every integer 2 \leq i \leq 10, \cos(\angle A_iA_1A_{i+1})=\frac{12}{13},

• The perimeter of the 11-gon A_1A_2\ldots A_{11} is equal to 20.

Then A_1A_2+A_1A_{11} can be expressed as \frac{m\sqrt{n}-p}{q} where m,n,p,q are positive integers, n is not divisible by any square, and no prime divides all of m,p, and q. Find m+n+p+q.

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.