Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer, and consider a sequence
![a_1, a_2, \ldots, a_n](/media/m/9/2/c/92c14c25a50ea2e6e7d3f457e8ea9a16.png)
of positive integers. Extend it periodically to an infinite sequence
![a_1, a_2, \ldots](/media/m/7/0/a/70a1247a1ab99045cbfc0d28aca8e204.png)
by defining
![a_{n+i} = a_i](/media/m/a/e/5/ae5decdf064e0877e7239ab734969d74.png)
for all
![i \geq 1](/media/m/4/8/2/482e79ab9501423a7bdfe6b5cce38027.png)
. If
![a_1 \leq a_2 \leq \cdots \leq a_n \leq a_1 + n](/media/m/0/a/4/0a44d849c316a332ff1128d1acc30e24.png)
and
![a_{a_i} \leq n + i - 1 \quad \quad \text{for}\ i = 1, 2, \ldots, n \text{,}](/media/m/c/4/d/c4d11bc5447c9fc7dbdc81f907aaf5f9.png)
prove that
%V0
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 $i \geq 1$. If $$
a_1 \leq a_2 \leq \cdots \leq a_n \leq a_1 + n
$$ and $$
a_{a_i} \leq n + i - 1 \quad \quad \text{for}\ i = 1, 2, \ldots, n \text{,}
$$ prove that $$
a_i + \cdots + a_n \leq n^2 \text{.}
$$