AT-OEMO-REG 2007 Problem 2

Find all tuples $(x_1,x_2,x_3,x_4,x_5)$ of positive integers with $x_1>x_2>x_3>x_4>x_5>0$ and ${\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.$