The problem has also been used in a 2007 Romanian Selection Test. Its origin is a submission by Serbia, from the 2007 Balkan MO Shortlist. One can prove that for integer $p\geq 1$, the function $f\colon \mathbb{N} \to \mathbb{Z}$ given by $f_p(n) = n^p - n!$ is injective, with the only exception $f_1(1)=f_1(2)$. The proof is quite arduous, and can be found in the Romanian Mathematical Competitions RMC 2007 brochure.

Thanks for the reference! I agree that in the general case the proof is arduous, but when $p=2007$ it is rather short and pleasant!