  We study the hadron spectra in nearly central $A$+$A$ collisions at RHIC and
LHC in a broad transverse momentum range. We cover the low-$p_T$ spectra using
longitudinally boost-invariant hydrodynamics with initial energy and net-baryon
number densities from the perturbative QCD (pQCD)+saturation model. Build-up of
the transverse flow and sensitivity of the spectra to a single decoupling
temperature $\Tdec$ are studied. Comparison with RHIC data at $\ssNN=130$ and
200 GeV suggests a rather high value $\Tdec=150$ MeV. The high-$p_T$ 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+$\bar{\rm p}$(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 $p_T$ spectra of hadrons in 5 % most central
Pb+Pb collisions at the LHC energy $\ssNN=5500$ GeV. Due to 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 $p_T$
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 $p_T$-region of applicability for hydrodynamical models at the LHC than
at RHIC.
