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We present the high-energy emission properties of GRB 160509A, from its prompt mission to late afterglow phase. GRB 160509A contains two emission episodes: 0-40s and 280-420s after the burst onset (t0). The relatively high fluence of GRB 160509A allows us to establish an evolving spectrum above 100 MeV. During the first emission episode, the >100 MeV spectrum is soft with \Gamma=>3.0, which can be smoothly connected to keV energies with a Band function with a high-energy cutoff. The >100 MeV spectrum rapidly changes to a hard spectrum with \Gamma<=1.5 after t0+40s. The existence of very energetic photons, e.g., a 52 GeV that arrives t0+77 seconds, and a 29 GeV that arrives t0+70 ks, is hard to reconcile by the synchrotron emission from forward-shock electrons, but likely due to inverse Compton mechanism (e.g., synchrotron self-Compton emission). A soft spectrum (\Gamma~2) between 300s and 1000s after the burst onset is also found at a significance of about 2 standard deviations, which suggests a different emission mechanism at work for this short period of time. GRB 160509A represents the latest example where inverse Compton emission has to be taken into account in explaining the afterglow GeV emission, which had been suggested long before the launch of Fermi LAT.