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A search for the decays $B_c^+ \to J/\psi D^{(*)0} K^+$ and $B_c^+ \to J/\psiD^{(*)+} K^{*0}$ is performed with data collected at the LHCb experimentcorresponding to an integrated luminosity of 3 fb$^{-1}$. The decays $B_c^+ \toJ/\psi D^0 K^+$ and $B_c^+ \to J/\psi D^{*0} K^+$ are observed for the firsttime, while first evidence is reported for the $B_c^+ \to J/\psi D^{*+} K^{*0}$and $B_c^+ \to J/\psi D^+ K^{*0}$ decays. The branching fractions of thesedecays are determined relative to the $B_c^+ \to J/\psi \pi^+$ decay. The$B_c^+$ mass is measured, using the $J/\psi D^0 K^+$ final state, to be$6274.28 \pm 1.40 (stat) \pm 0.32 (syst)$ MeV/$c^2$. This is the most precisesingle measurement of the $B_c^+$ mass to date.

Observation of $B_{c}^{+} \to J/\psi D^{(*)} K^{(*)}$ decays

Observation of $B_{c}^{+} \to J/\psi D^{()} K^{()}$ decays