  We study the exotic $bc\bar{q}\bar{q}$, $bc\bar{s}\bar{s}$ and
$qc\bar{q}\bar{b}$, $sc\bar{s}\bar{b}$ systems by constructing the
corresponding tetraquark currents with $J^P=0^+$ and $1^+$. After investigating
the two-point correlation functions and the spectral densities, we perform QCD
sum rule analysis and extract the masses of these open-flavor tetraquark
states. Our results indicate that the masses of both the scalar and axial
vector tetraquark states are about $7.1-7.2$ GeV for the $bc\bar{q}\bar{q}$
system and $7.2-7.3$ GeV for the $bc\bar{s}\bar{s}$ system. For the
$qc\bar{q}\bar{b}$ tetraquark states with $J^P=0^+$ and $1^+$, their masses are
extracted to be around $7.1$ GeV. The masses for the scalar and axial vector
$sc\bar{s}\bar{b}$ states are $7.1$ GeV and $6.9-7.1$ GeV, respectively. The
tetraquark states $qc\bar{q}\bar{b}$ and $sc\bar{s}\bar{b}$ lie below the
thresholds of $D^{(\ast)}B^{(\ast)}$ and $D_s^{(\ast)}B_s^{(\ast)}$
respectively, but they can decay into $B_c$ plus a light meson. However, the
tetraquark states $bc\bar{q}\bar{q}$ and $bc\bar{s}\bar{s}$ lie below the
$D^{(\ast)}\bar B^{(\ast)}$ and $D_s^{(\ast)}\bar B_s^{(\ast)}$ thresholds,
suggesting dominantly weak decay mechanisms.
