It is well known (cf. Krajíček and Pudlák [‘Propositional proof systems, the consistency of first order theories and the complexity of computations’, J. Symbolic Logic 54 (1989) 1063–1079]) that a polynomial time algorithm finding tautologies hard for a propositional proof system P exists if and only if P is not optimal. Such an algorithm takes 1(k) and outputs a tautology τk of size at least k such that P is not p-bounded on the set of all formulas τk. We consider two more general search problems involving finding a hard formula, Cert and Find, motivated by two hypothetical situations: that one can prove that NP≠coNP and that no optimal proof system exists. In Cert one is asked to find a witness that a given non-deterministic circuit with k inputs does not define TAUT∩{0, 1}k. In Find, given 1(k) and a tautology α of size at most k c 0 , one should output a size k tautology β that has no size k c 1 P-proof from substitution instances of α. We will prove, assuming the existence of an exponentially hard one-way permutation, that Cert cannot be solved by a time 2O(k) algorithm. Using a stronger hypothesis about the proof complexity of the Nisan–Wigderson generator, we show that both problems Cert and Find are actually only partially defined for infinitely many k (that is, there are inputs corresponding to k for which the problem has no solution). The results are based on interpreting the Nisan–Wigderson generator as a proof system.