Determine all possible quintuplets (p, q, r, s, t) of positive integers that satisfy this equation:

⌊(p+q)/3⌋² +⌊(q+r)/3⌋² + ⌊(r+s)/3⌋² + ⌊(s+t)/3⌋² = 38

*** ⌊p⌋ is the floor function. ie. the greatest integer less than or equal to p.

****** Adapted from a problem appearing at an Austrian Mathematical Olympiad in 2007.

**** Computer program assisted solutions are welcome, but a semi-analytic methodology, that is: hand calculator and p&p, is preferred.