  Three fermions with strongly repulsive interactions in a spherical harmonic
trap, constitute the simplest nontrivial system that can exhibit the onset of
itinerant ferromagnetism. Here, we present exact solutions for three trapped,
attractively interacting fermions near a Feshbach resonance. We analyze energy
levels on the upper branch of the resonance where the atomic interaction is
effectively repulsive. When the s-wave scattering length a is sufficiently
positive, three fully polarized fermions are energetically stable against a
single spin-flip, indicating the possibility of itinerant ferromagnetism, as
inferred in the recent experiment. We also investigate the high-temperature
thermodynamics of a strongly repulsive or attractive Fermi gas using a quantum
virial expansion. The second and third virial coefficients are calculated. The
resulting equations of state can be tested in future quantitative experimental
measurements at high temperatures and can provide a useful benchmark for
quantum Monte Carlo simulations.
