Within the framework of QCD sum rules, we have investigated the tetraquark states with three heavy quarks. We systematically construct the interpolating currents for the possible $ccbar{c}bar{q}$, $ccbar{b}bar{q}$, $bcbar{b}bar{q}$, $bbbar{b}bar{q}$ tetraquark states with quantum numbers $J^{P}=0^{+}$ and $J^{P}=1^{+}$. Using these interpolating currents, we have calculated the two-point correlation functions and extracted the mass spectra for the above tetraquark states. We also discuss the decay patterns of these tetraquarks, and notice that the $ccbar{c}bar{q}$, $ccbar{b}bar{q}$, $bcbar{b}bar{q}$ may decay quickly with a narrow width due to their mass spectra. The $bbbar{b}bar{q}$ tetraquarks are expected to be very narrow resonances since their OZI-allowed decay modes are kinematically forbidden. These states may be searched for in the final states with a $B$ meson plus a light meson or photon.
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