We introduce a new color-selection technique to identify high-redshift, massive galaxies that are systematically missed by Lyman-break selection. The new selection is based on the H_{160} and IRAC 4.5um bands, specifically H - [4.5] > 2.25 mag. These galaxies, dubbed HIEROs, include two major populations that can be separated with an additional J - H color. The populations are massive and dusty star-forming galaxies at z > 3 (JH-blue) and extremely dusty galaxies at z < 3 (JH-red). The 350 arcmin^2 of the GOODS-N and GOODS-S fields with the deepest HST/WFC3 and IRAC data contain 285 HIEROs down to [4.5] < 24 mag. We focus here primarily on JH-blue (z > 3) HIEROs, which have a median photometric redshift z ~4.4 and stellar massM_{*}~10^{10.6} Msun, and are much fainter in the rest-frame UV than similarly massive Lyman-break galaxies (LBGs). Their star formation rates (SFRs) reaches ~240 Msun yr^{-1} leading to a specific SFR, sSFR ~4.2 Gyr^{-1}, suggesting that the sSFRs for massive galaxies continue to grow at z > 2 but at a lower growth rate than from z=0 to z=2. With a median half-light radius of 2 kpc, including ~20% as compact as quiescent galaxies at similar redshifts, JH-blue HIEROs represent perfect star-forming progenitors of the most massive (M_{*} > 10^{11.2} Msun) compact quiescent galaxies at z ~ 3 and have the right number density. HIEROs make up ~60% of all galaxies with M_{*} > 10^{10.5} Msun identified at z > 3 from their photometric redshifts. This is five times more than LBGs with nearly no overlap between the two populations. While HIEROs make up 15-25% of the total SFR density at z ~ 4-5, they completely dominate the SFR density taking place in M_{*} >10^{10.5} Msun galaxies, and are therefore crucial to understanding the very early phase of massive galaxy formation.
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