Problem

Source: Polish Mathematical Olympiad finals 2021 P2

Tags: function, Functional inequality



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