VN-TST 2014 Problem 6

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$ D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$ . A filling of integers is called Dien Bien if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$ , the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many Dien Bien fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation.