  We theoretically investigate ground-state properties of a three-component
Fermi gas with pairwise contact interactions between different components near
a triatomic resonance where bound trimers are about to appear. Using
variational equations for in-medium two- and three-body cluster states in three
dimensions, we elucidate the competition of pair and triple formations due to
the Fermi surface effects. We present the ground-state phase diagram that
exhibits transition from a Cooper pair to Cooper triple state and crossover
from a Cooper triple to tightly bound trimer state at negative scattering
lengths. This three-body crossover is analogous to the
Bardeen-Cooper-Schrieffer to Bose-Einstein condensation crossover observed in a
two-component Fermi gas. We predict that the threshold scattering length
$a_{-}$ for three-body states can be shifted towards the weak-coupling side due
to the emergence of Cooper triples.
