We present a 1.1~mm census of dense cores in the Mon~R2 Giant Molecular Cloud with the AzTEC instrument on the Large Millimeter Telescope (LMT). We detect 295 cores (209 starless, and 86 with protostars) in a two square degree shallow survey. We also carry out a deep follow-up survey of 9 regions with low to intermediate ($3<A_V<7$) gas column densities and detect 60 new cores in the deeper survey which allows us to derive a completeness limit. After performing corrections for low signal-to-noise cores, we find a median core mass of $sim 2.1 text{M}_{odot}$ and a median size of $ 0.08$~pc. $46%$ of the cores (141) have masses exceeding the local Bonor-Ebert mass for cores with T=12K, suggesting that in the absence of supporting non-thermal pressure, these regions are unstable to gravitational collapse. We present the core mass function (CMF) for various subdivisions of the core sample. We find that cores with masses $>$10~$M_{odot}$ are exclusively found in regions with high core number densities and that the CMF of the starless cores has an excess of low-mass cores ($<$5~$M_{odot}$) compared to the CMF of protostellar cores. We report a power law correlation of index $1.99 pm 0.03$ between local core mass density and gas column density (as traced by Herschel) over a wide range of size scales (0.3-5~pc). This power law is consistent with that predicted for thermal fragmentation of a self-gravitating sheet.
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