  Tian's criterion for K-stability states that a Fano variety of dimension $n$
whose alpha invariant is greater than $\frac{n}{n+1}$ is K-stable. We show that
this criterion is sharp by constructing singular Fano varieties with alpha
invariants $\frac{n}{n+1}$ that are not K-polystable for sufficiently large
$n$. We also construct K-unstable Fano varieties with alpha invariants
$\frac{n-1}{n}$.
