  We establish various general results concerning static and spherically
symmetric black hole solutions of general higher-derivative extensions of
Einstein gravity. We prove that the only theories susceptible of admitting
solutions with $g_{tt}g_{rr}=-1$ and representing the exterior field of a
spherically symmetric distribution of mass are those that only propagate a
massless and traceless graviton on the vacuum. Then, we provide a simple (and
computationally powerful) sufficient condition for a theory to admit solutions
of that kind, as well as a systematic way for constructing them for a given
theory. We conjecture (and provide strong evidence) that all black holes
constructed according to our criteria are completely determined by their mass
(non-hairy), and such that their thermodynamic properties can be obtained by
solving a system of algebraic equations without free parameters. Our results
can be straightforwardly extended to planar and hyperbolic horizons. We
illustrate this by obtaining new planar asymptotically $AdS_5$ black hole
solutions of the recently constructed Generalized quasitopological gravity
[arXiv:1703.01631], which belongs to the class of theories selected by our
results.
