We study the hadron spectra in nearly central A+A collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) in a broad transverse momentum range. We cover the low-pT spectra using longitudinally boost-invariant hydrodynamics with initial energy and net-baryon number densities from the perturbative QCD (pQCD)+saturation model. Buildup of the transverse flow and sensitivity of the spectra to a single decoupling temperature Tdec are studied. Comparison with RHIC data at sqrt[sNN]=130 and 200 GeV suggests a rather high value Tdec=150 MeV. The high-pT spectra are computed using factorized pQCD cross sections, nuclear parton distributions, fragmentation functions, and describing partonic energy loss in the quark-gluon plasma by quenching weights. Overall normalization is fixed on the basis of p+p̅ (p) data and the strength of energy loss is determined from RHIC Au+Au data. Uncertainties are discussed. With constraints from RHIC data, we predict the pT spectra of hadrons in 5% most central Pb+Pb collisions at the LHC energy sqrt[sNN]=5500 GeV. Because of the closed framework for primary production, we can also predict the net-baryon number at midrapidity, as well as the strength of partonic energy losses at the LHC. Both at the LHC and RHIC, we recognize a rather narrow crossover region in the pT spectra, where the hydrodynamic and pQCD fragmentation components become of equal size. We argue that in this crossover region the two contributions are to a good approximation mutually independent. In particular, our results suggest a wider pT region of applicability for hydrodynamical models at the LHC than at RHIC.