This is a Regional Mathematics Olympiad, RMO 2015 Problem 3 from West Bengal Region.
Actually, I had a typographical mistake in my last post i.e. question paper.
The 3rd post says to find integer solutions, not positive integer solutions.
I consider this problem as the easiest problem in RMO 2015. So let's discuss the solution...
Source :- RMO 2015 West Bengal Problem 3
Show that there are infinitely many triplets of integers such that
Let us put We get
Now, let be 2 integers, such that
If we put then we have
Which is true.
Hence, every triplet of the forms, where are integers, such that is a solution to the equation.
And, as takes infinitely many values, the equation has infinitely many solutions.
This completes the proof.