MathNet 2012 Question 91

Suppose there are beetles on a chessboard consisting of $2012 \times 2012$ unit squares. Each unit square can accommodate at most one beetle. At a moment, all beetles fly and land on the chessboard again. For a beetle, we call the vector from its flying unit to its landing unit the beetle's "displacement vector". We call the sum of all beetle's "displacement vectors" the "total displacement vectors".
Find the maximum length of "total displacement vector" considering the number of beetles and all possible positions of flying and landing. (posed by Qu Zhenhua)