2003 AIME II Problem 14

Let A = (0,0) and B = (b,2) be points on the coordinate plane. Let ABCDEF be a convex equilateral hexagon such that \angle FAB = 120^\circ, \overline{AB}\parallel \overline{DE}, \overline{BC}\parallel \overline{EF,} \overline{CD}\parallel \overline{FA}, and the y-coordinates of its vertices are distinct elements of the set \{0,2,4,6,8,10\}. The area of the hexagon can be written in the form m\sqrt {n}, where m and n are positive integers and n is not divisible by the square of any prime. Find m + n.

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.