The Examiner's Problem (uniform generation of SAT decision instances/answers)
The question you are asking is equivalent to unary NP = unary P, which in turn is equivalent to NE = E, by padding.

From the title, perhaps you meant to ask if it is possible to generate input/output pairs such that the distribution on the inputs is "hard."  The possibility of doing this lies somewhere between P $\ne$ NP and one-way functions exist.

In restricted computational models, it is known that this is possible.  E.g. one can generate input/output pairs for the parity or majority functions in AC$^0$ or below.  See The complexity of distributions.