FKMO 2017 Problem 3

For a positive integer $n$ , denote $c_n=2017^n$ . A function $f: \mathbb{N} \rightarrow \mathbb{R}$ satisfies the following two conditions.

1. For all positive integers $m, n$ , $f(m+n) \le 2017 \cdot f(m) \cdot f(n+325)$ .

2. For all positive integer $n$ , we have $0<f(c_{n+1})<f(c_n)^{2017}$ .

Prove that there exists a sequence $a_1, a_2, \cdots$ which satisfies the following. For all $n, k$ which satisfies $a_k<n$ , we have $f(n)^{c_k} < f(c_k)^n$ .