We present the results of our ALMA HCN J=3-2 and HCO+ J=3-2 line observations of a uniformly selected sample (>25) of nearby ultraluminous infrared galaxies (ULIRGs) at z < 0.15. The emission of these dense molecular gas tracers and continuum are spatially resolved in the majority of observed ULIRGs for the first time with achieved synthesized beam sizes of ~0.2 arcsec or ~500 pc. In most ULIRGs, the HCN-to-HCO+ J=3-2 flux ratios in the nuclear regions within the beam size are systematically higher than those in the spatially extended regions. The elevated nuclear HCN J=3-2 emission could be related to (a) luminous buried active galactic nuclei, (b) the high molecular gas density and temperature in ULIRGs nuclei, and/or (c) mechanical heating by spatially compact nuclear outflows. A small fraction of the observed ULIRGs display higher HCN-to-HCO+ J=3-2 flux ratios in localized off-nuclear regions than those of the nuclei, which may be due to mechanical heating by spatially extended outflows. The observed nearby ULIRGs are generally rich in dense (>10^5 cm^-3) molecular gas, with an estimated mass of >10^9 Msun within the nuclear (a few kpc) regions, and dense gas can dominate the total molecular mass there. We find a low detection rate (<20%) regarding the possible signature of a vibrationally excited (v2=1f) HCN J=3-2 emission line in the vicinity of the bright HCO+ J=3-2 line that may be due, in part, to the large molecular line widths of ULIRGs.
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