Nonnegative integers $x_1, x_2 , \ldots, x_n$ are such that $x_1^2 + x_2^2 + \ldots +x_n^2 = n$, $n > 1$. Prove that for any nonnegative real numbers $y_1, y_2 , \ldots, y_n$ there exist indices $i, j \in \{1, 2, \ldots, n\}$, not necessarily different, for which the following inequality holds: $$x_i + y_j - x_{j+1}y_{i+1} \geq 1$$ (Here index $n+1 = $ index $1$). Proposed by Nazar Serdyuk