  The experimentally known $B_c$ states are all below open bottom-charm
threshold, which experience three main decay modes, and all induced by weak
interaction. In this work, we investigate the mass spectrum and strong decays
of the $B_c(3S)$ states, which just above the threshold, in the Bethe-Salpeter
formalism and $^3P_0$ model. The numerical estimation gives
$M(B_c(3^1S_0))=7273\ {\rm MeV}$, $M(B_c^*(3^3S_1))=7304\ {\rm MeV}$,
$\Gamma\left(B_c(3^1S_0)\to B^*D\right)=26.02^{+2.33}_{-2.21}\ {\rm MeV}$,
$\Gamma\left(B_c^*(3^3S_1)\to BD\right)=3.39^{+0.27}_{-0.26}\ {\rm MeV}$,
$\Gamma\left(B_c^*(3^3S_1)\to B^*D\right)=14.77^{+1.40}_{-1.33}\ {\rm MeV}$ and
$\Gamma\left(B_c^*(3^3S_1)\to BD^*\right)=6.14^{+0.58}_{-0.54}\ {\rm MeV}$.
Compared with previous studies in non-relativistic approximation, our results
indicate that the relativistic effects are notable in $B_c(3S)$ exclusive
strong decays. According to the results, we suggest to find the $B_c(3S)$
states in their hadronic decays to $B$ and $D$ mesons in experiment, like the
LHCb.
