MathDB

A cat is tied to one corner of the base of a tower. The base forms an equilateral triangle of side length

4

m, and the cat is tied with a leash of length

8

m. Let

A

be the area of the region accessible to the cat. If we write

A = \frac{m}{n} k - \sqrt{\ell}

, where

m,n, k, \ell

are positive integers such that

m

and

n

are relatively prime, and

\ell

is squarefree, what is the value of

m + n + k + \ell

2