We study the exotic $bc\overline{q}\overline{q}$, $bc\overline{s}\overline{s}$ and $qc\overline{q}\overline{b}$, $sc\overline{s}\overline{b}$ 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 $bc\overline{q}\overline{q}$ system and 7.2–7.3 GeV for the $bc\overline{s}\overline{s}$ system. For the $qc\overline{q}\overline{b}$ 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 $sc\overline{s}\overline{b}$ states are 7.1 and 6.9–7.1 GeV, respectively. The tetraquark states $qc\overline{q}\overline{b}$ and $sc\overline{s}\overline{b}$ lie below the thresholds of $D^{(*)}{B}^{(*)}$ and $D_s^{(*)}{B}_s^{(*)}$ respectively, but they can decay into $B_c$ plus a light meson. However, the tetraquark states $bc\overline{q}\overline{q}$ and $bc\overline{s}\overline{s}$ lie below the $D^{(*)}{\overline{B}}^{(*)}$ and $D_s^{(*)}{\overline{B}}_s^{(*)}$ thresholds, suggesting dominantly weak decay mechanisms.

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