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Searches are presented for Bc+→Dh+h- decays, where D is a charmed meson and h± is a charged pion or kaon, using pp collision data collected by the LHCb experiment corresponding to an integrated luminosity of 9 fb-1. The decays Bc+→D+K+π-, Bc+→D∗+K+π-, and Bc+→Ds+K+K- are observed for the first time. Their branching fractions, expressed as ratios relative to that of the Bc+→Bs0π+ decay, are determined to be R(Bc+→D+K+π-)=(1.96±0.23±0.08±0.10)×10-3, R(Bc+→D∗+K+π-)=(3.67±0.55±0.24±0.20)×10-3, R(Bc+→Ds+K+K-)=(1.61±0.35±0.13±0.07)×10-3, where the first uncertainty is statistical, the second is systematic, and the third is due to the limited precision on the D-meson branching fractions. The decay channels proceed primarily through excited K0 or D0 resonances or φ mesons, and open a new avenue for studies of charge-parity violation in beauty mesons.