2003 AIME I Problem 6

The sum of the areas of all triangles whose vertices are also vertices of a 1 by 1 by 1 cube is m + \sqrt{n} + \sqrt{p}, where m, n, and p are integers. Find m + n + p.

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.