Omni-MATH / Consider pairs (f,g) of functions from the set of nonnegative integers to itself such that [list] [*]f(0)≥f(1)≥f(2)≥…≥f(300)≥0 [*]f(0)+f(1)+f(2)+…+f(300)≤300 [*]for any 20 nonnegat…

Consider pairs (f,g) of functions from the set of nonnegative integers to itself such that 
[list]
[*]f(0)≥f(1)≥f(2)≥…≥f(300)≥0
[*]f(0)+f(1)+f(2)+…+f(300)≤300
[*]for any 20 nonnegative integers n₁,n₂,…,n₂₀, not necessarily distinct, we have g(n₁+n₂+…+n₂₀)≤f(n₁)+f(n₂)+…+f(n₂₀).
[/list]
Determine the maximum possible value of g(0)+g(1)+…+g(6000) over all such pairs of functions.

[i]Sean Li[/i]

Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that 
[list]
[*]$f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0$
[*]$f(0)+f(1)+f(2)+\dots+f(300) \leq 300$
[*]for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have $$g(n_1+n_2+\dots+n_{20}) \leq f(n_1)+f(n_2)+\dots+f(n_{20}).$$
[/list]
Determine the maximum possible value of $g(0)+g(1)+\dots+g(6000)$ over all such pairs of functions.

[i]Sean Li[/i]