Question 4
Let $n$ be a positive integer, and consider a sequence $a_{1}, a_{2}, \ldots, a_{n}$ of positive integers. Extend it periodically to an infinite sequence $a_{1}, a_{2}, \ldots$ by defining $a_{n+i}=a_{i}$ for all $i \geqslant 1$. If
$$
a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n} \leqslant a_{1}+n
$$
and
$$
a_{a_{i}} \leqslant n+i-1 \quad \text { for } i=1,2, \ldots, n
$$
prove that
$$
a_{1}+\cdots+a_{n} \leqslant n^{2} .
$$
$$
a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n} \leqslant a_{1}+n
$$
and
$$
a_{a_{i}} \leqslant n+i-1 \quad \text { for } i=1,2, \ldots, n
$$
prove that
$$
a_{1}+\cdots+a_{n} \leqslant n^{2} .
$$