> You must assume that at least some of the coefficients of the x terms are non-zero.
That turns out not to be the case.
>> Do you not find that surprising?
>> how would you explain it?
> I think it is up to you to explain why this is interesting.
So far, many people who are clearly intelligent seem to miss that it works at all, let alone how it works. You appear so far to be among them. As I've said elsewhere, I appreciate that you are trying to poke holes in it, but it really is complete and correct as stated.
> Without the condition that k is greater than all the coefficients, the solution is clearly non-unique.
Well, yes. That's why having k greater than all the coefficients is a requirement. That's why your example in another comment elsewhere is not an example of the puzzle failing - it does not satisfy the condition that k must exceed all the coefficients.
> Given that, the assumption of the coefficients are integers, and that the solution is not a constant polynomial, the statement becomes equivalent to "division is a function".
I don't see why this is true - perhaps you could expand on it? Why is the problem statement equivalent to saying "Division is a function."?
So now I don't understand what you're trying to say or do. The question as stated is fine - an example that doesn't satisfy the conditions isn't a counter-example. So at least one of us is confused.
So let me state clearly, rather than having a bitty back'n'forth.
You pick a polynomial p(x) with non-negative integer coefficients. You choose an integer k that's larger than all the coefficients of p(x). You tell me k and p(k). In return, I tell you p(x).
Are you claiming this doesn't work?
Are you claiming it's trivial to accomplish?
Are you claiming it's obvious why it works?
I'm no longer sure what you're claiming, so I thought I'd ask clearly and plainly.