For the system \begin{equation} u''_i=f_i(t,u_1,\dots,u_n,u_1',\dots,u_n')\quad (i=1,\dots,n) \tag{1} \end{equation} a periodic solution exists if for each i one of the following inequalities holds: \begin{gather*} f_i(t,u_1,\dots,u_n,p_1,\dots,p_n)\leqslant A(p_1,\dots,p_{i-1})p_i^2+B(p_1,\dots,p_{i-1}), \\ f_i(t,u_1,\dots,u_n,p_1,\dots,p_n)\geqslant-A(p_1,\dots,p_{i-1})p_i^2-B(p_1,\dots,p_{i-1}), \\ f_i(t,u_1,\dots,u_n,p_1,\dots,p_n)\operatorname{sign}\leqslant A(p_1,\dots,p_{i-1})p_i^2+B(p_1,\dots,p_{i-1}), \\ f_i(t,u_1,\dots,u_n,p_1,\dots,p_n)\operatorname{sign}\geqslant-A(p_1,\dots,p_{i-1})p_i^2-B(p_1,\dots,p_{i-1}), \\ f_i(t,u_1,\dots,u_n,p_1,\dots,p_n)\operatorname{sign}u_i\geqslant-A(p_1,\dots,p_{i-1})p_i^2-B(p_1,\dots,p_{i-1}), \end{gather*} for $\alpha(t)\leqslant u\leqslant\beta(t)$. Here $\alpha(t)$ and $\beta(t)$ are the lower and upper vector functions for system (1) and the periodic conditions; $A\geqslant0$, $B\geqslant0$. Bibliography: 1 titles.