Question 20
Given a natural number $n$ and natural numbers $a_1,\ldots,a_n$, extend the sequence periodically by $a_{n+i}=a_i$ for all $i\ge1$. Suppose:
(i) $a_1\le a_2\le\cdots\le a_n\le a_1+n$.
(ii) $a_{a_i}\le n+i-1$ for $i=1,2,\ldots,n$.
Prove $a_1+a_2+\cdots+a_n\le n^2$.
(i) $a_1\le a_2\le\cdots\le a_n\le a_1+n$.
(ii) $a_{a_i}\le n+i-1$ for $i=1,2,\ldots,n$.
Prove $a_1+a_2+\cdots+a_n\le n^2$.