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Introducing superconductivity (SC) or magnetism into topological insulators (TIs) can give rise to novel quantum states and exotic physical phenomena. Here, we report a high-pressure transport study on the TI Ge2Bi2Te5 and its Mn-doped counterparts. The application of pressure induces a SC in Ge2Bi2Te5, which shows a dome-shape phase diagram with the maximum Tc of 7.6 K at 23 GPa. Doping Mn into Ge2Bi2Te5 introduces an antiferromagnetic order at ambient pressure and strongly weakens the pressure-induced SC, demonstrating that magnetism and SC compete in this material system. Present study provides a new platform for investigating the interplay among band topology, magnetism, and SC.

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