Question

how many solutions does this problem have?

Answer 1

no problem = no solution....

Answer 2

Touche.

Answer 3

if this is a logic question, then I would say that it has exactly one solution.

take the case where it had no solutions, then you would write "it has no solutions"
take the case where it had 1 solution, then you would write "it has one solution"
take the case where it had 2 solutions, then you would write "it has 2 solutions"
...
take the case where it had N solutions, then you would write "it has N solutions"

in ALL cases, you wrote out one answer, hence it has exactly one solution.