Below, we present a problem from the 2/25 Algebra and Number Theory set, along with 3 student submitted solution. You may vote up for the solutions that you think should be featured, and should vote down for those solutions that you think are wrong.
Integrally rational How many ordered triples of positive integers with are there such that has a rational solution?
You may try the problem by clicking on the above link.
All solutions may have LaTeX edits to make the math appear properly. The exposition is presented as is, and has not been edited.
If you think that all these solutions are essentially the same, read VERY carefully. They differ in one important area. Most submitted solutions were not marked correct.
There are 2 parts to this question. The first part involves arguing that a solution occurs if and only if is a perfect square, and the second part involves actually counting the number of possibilities. Most students did not do the first part well.
Solution A - This solution didn't do the second part. If you look at his logical implications, he only showed that "If has a rational number", then " is the square of an integer". He then proceeds to state that the number of cases where is 13. This doesn't answer the original question, and merely shows that the answer is at most 13.
Solution B - This logical deductions in this solution does not hold. It is not true that " (A) To make has a rational solution", "(B) must be an integer", "(C) or must be a perfect square. In fact, all that (A) implies is " must be a rational number". We then need to show that "since is an integer, hence is a perfect square". Likewise, this does not explain why those 13 cases satisfy the original conditions.
Solution C - This statement of " one of these numbers is rational if and only if " is the only correct solution. It has an error in Case 5 as pointed out by Bob. This solution is presented by Sreejato.