We present new quasars discovered in the vicinity of the Andromeda and Triangulum galaxies with the LAMOST during the 2010 and 2011 observational seasons. Quasar candidates are selected based on the available SDSS, KPNO 4 m telescope, XSTPS optical, and WISE near infrared photometric data. We present 509 new quasars discovered in a stripe of ~135 sq. deg from M31 to M33 along the Giant Stellar Stream in the 2011 pilot survey datasets, and also 17 new quasars discovered in an area of ~100 sq. deg that covers the central region and the southeastern halo of M31 in the 2010 commissioning datasets. These 526 new quasars have i magnitudes ranging from 15.5 to 20.0, redshifts from 0.1 to 3.2. They represent a significant increase of the number of identified quasars in the vicinity of M31 and M33. There are now 26, 62 and 139 known quasars in this region of the sky with i magnitudes brighter than 17.0, 17.5 and 18.0 respectively, of which 5, 20 and 75 are newly-discovered. These bright quasars provide an invaluable collection with which to probe the kinematics and chemistry of the ISM/IGM in the Local Group of galaxies. A total of 93 quasars are now known with locations within 2.5 deg of M31, of which 73 are newly discovered. Tens of quasars are now known to be located behind the Giant Stellar Stream, and hundreds behind the extended halo and its associated substructures of M31. The much enlarged sample of known quasars in the vicinity of M31 and M33 can potentially be utilized to construct a perfect astrometric reference frame to measure the minute PMs of M31 and M33, along with the PMs of substructures associated with the Local Group of galaxies. Those PMs are some of the most fundamental properties of the Local Group.
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