Unimprovable, in a certain sense, sufficient conditions of solvability and unsolvability of nonlocal problems are found for the differential system
$$ \frac{dx_i}{dt} =f_i(t,x_1,\dots,x_n) \quad (i=1,\dots,n), $$
where each of the functions $f_i:[a,b]\times R^n \to R$ $(i=1,\dots,n)$ may be superlinear or sublinear with respect to phase variables.
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