PL-OM 2021 Problem 2

Let $n$ be an integer. For pair of integers $0 \leq i,$ $j\leq n$ there exist real number $f(i,j)$ such that:

1. $f(i,i)=0$ for all integers $0\leq i \leq n$ 2) $0\leq f(i,l) \leq 2\max \{ f(i,j), f(j,k), f(k,l) \}$ for all integers $i$ , $j$ , $k$ , $l$ satisfying $0\leq i\leq j\leq k\leq l\leq n$ .

Prove that $$f(0,n) \leq 2\sum_{k=1}^{n}f(k-1,k)$$