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Observations of exotic structures in the J/psi p channel, which we refer to as charmonium-pentaquark states, in Lambda(0)(b) --> J/psi K(-)p decays are presented. The data sample corresponds to an integrated luminosity of 3 fb(-1) acquired with the LHCb detector from 7 and 8 TeV pp collisions. An amplitude analysis of the three-body final state reproduces the two-body mass and angular distributions. To obtain a satisfactory fit of the structures seen in the J/psi p mass spectrum, it is necessary to include two Breit-Wigner amplitudes that each describe a resonant state. The significance of each of these resonances is more than 9 standard deviations. One has a mass of 4380 +/- 8 +/- 29 MeV and a width of 205 +/- 18 +/- 86 MeV, while the second is narrower, with a mass of 4449.8 +/- 1.7 +/- 2.5 MeV and a width of 39 +/- 5 +/- 19 MeV. The preferred J(P) assignments are of opposite parity, with one state having spin 3/2 and the other 5/2.
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