  We provide a sufficient condition for polarisations of Fano varieties to be
K-stable in terms of Tian's alpha invariant, which uses the log canonical
threshold to measure singularities of divisors in the linear system associated
to the polarisation. This generalises a result of Odaka-Sano in the
anti-canonically polarised case, which is the algebraic counterpart of Tian's
analytic criterion implying the existence of a K\"{a}hler-Einstein metric. As
an application, we give new K-stable polarisations of a general degree one del
Pezzo surface. We also prove a corresponding result for log K-stability.
