= Solution
In the decelerating <hot Big Bang model>, the <particle horizon> at recombination encloses only a finite comoving region. Widely separated parts of the <cosmic microwave background> therefore had no overlapping past light cones in that model, although they have almost the same temperature. This is the <horizon problem>. During <cosmic inflation>, accelerated expansion makes the <comoving Hubble radius> $(aH)^{-1}$ shrink. A small region that was initially in causal contact can then expand to contain the whole presently observable universe.
The <flatness problem> follows from
$$
|\Omega-1|=\frac{|k|}{a^2H^2}.
$$
During ordinary radiation- or matter-dominated decelerating expansion this quantity grows, so the observed closeness of the <cosmological density parameter> $\Omega$ to one requires extreme tuning at early times. Inflation makes $aH$ grow nearly exponentially and therefore drives $|\Omega-1|$ rapidly toward zero. A sufficiently large <number of e-folds> makes spatial flatness a dynamical outcome rather than a finely tuned initial condition.
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