Question 15
The sides of rectangle $ABCD$ have lengths $10$ and $11.$ An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$ The maximum possible area of such a triangle can be written in the form $p\sqrt{q} - r,$ where $p,$ $q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p + q + r.$