Decays of (B)over bar(s)(0)and (B)over bar (0) mesons into J/psi pi(+)pi(-)pi(+)pi(-) final states, produced in pp collisions at the LHC, are investigated using data corresponding to an integrated luminosity of 3fb(-1) collected with the LHCb detector. (B)over bar((s))(0)?J/psi f(1)(1285) decays are seen for the first time, and the branching fractions are measured. Using these rates, the f(1)(1285) mixing angle between strange and nonstrange components of its wave function in the q(q)over bar structure model is determined to be +/-(24.0(-2.6-0.8)(+3.1+0.6))degrees. Implications on the possible tetraquark nature of the f(1)(1285) are discussed.