Kevin sent in his work on this question:

We know that Mona's number plus Bob's number is 5 less than a square and 6 more than a square. So these squares must be 11 apart. I wrote out the first few squares, and saw that they get further and further apart, and the only ones that are 11 apart are 25 and 36. So Mona's number plus Bob's number is 31.

We know that when you add them all up you get a square, so Jamie's number plus 31 is a square. From the hint, all the numbers are less than 40, so Jamie's number plus 31 is 36. So Jamie's number is 5.

Then I found that the only way we could make the rest of the problem work is to have Mona's number as 11 and Bob's as 20 (or the other way round).