  Many dark energy models fail to pass the cosmic age test. In this paper, we
investigate the cosmic age problem associated with nine extremely old Global
Clusters (GCs) and the old quasar APM $08279+5255$ in the $R_h=ct$ Universe.
The age data of these oldest GCs in M31 is acquired from the
Beijing-Arizona-Taiwan-Connecticut system with up-to-date theoretical synthesis
models. They have not been used to test the cosmic age problem in the $R_h=ct$
Universe in previous literature. By evaluating the age of the $R_h=ct$ Universe
with the observational constraints from the type Ia supernovae and Hubble
parameter, we find that the $R_h=ct$ Universe can accommodate five GCs and the
quasar APM 08279+5255 at redshift $z=3.91$. But for other models, such as
$\Lambda$CDM, interacting dark energy model, generalized Chaplygin gas model
and holographic dark energy model, can not accommodate all GCs and the quasar
APM 08279+5255. It is worthwhile to note that the age estimates of some GCs are
controversial. So, unlike other cosmological models, the $R_h=ct$ Universe can
marginally solve the cosmic age problem, especially at high redshift.
