Question 190
A positive integer $n$ and a function $f : \mathbb{N} \to \mathbb{N}$ are such that:
(1) $f(1) \le f(2) \le \dots \le f(n) \le f(1) + n$;
(2) $f(n + i) = f(i)$ for any positive integer $i$;
(3) $f(f(i)) \le n + i - 1$ for any positive integer $i$.
Prove that $f(1) + f(2) + \dots + f(n) \le n^2$.
(1) $f(1) \le f(2) \le \dots \le f(n) \le f(1) + n$;
(2) $f(n + i) = f(i)$ for any positive integer $i$;
(3) $f(f(i)) \le n + i - 1$ for any positive integer $i$.
Prove that $f(1) + f(2) + \dots + f(n) \le n^2$.