According the the introduction to Mazur's *Rational Isogenies of Prime Degree* the following question was open in 1978:

Let $N$ be one of the integers 39, 65, 91, 125, or 169. Does the modular curve $X_0(N)$ possess noncuspidal rational points?

It seems likely that this should have been resolved in the past 32 years. Does anyone know of a reference for this?