Question 174
Consider pairs $(f, g)$ of functions from the set of nonnegative integers to itself such that
* $f(0) \ge f(1) \ge f(2) \ge \dots \ge f(300) \ge 0$;
* $f(0) + f(1) + f(2) + \dots + f(300) \le 300$;
* for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have
$$
g(n_1 + n_2 + \dots + n_{20}) \le f(n_1) + f(n_2) + \dots + f(n_{20}).
$$
Determine the maximum possible value of $g(0) + g(1) + \dots + g(6000)$ over all such pairs of functions.
* $f(0) \ge f(1) \ge f(2) \ge \dots \ge f(300) \ge 0$;
* $f(0) + f(1) + f(2) + \dots + f(300) \le 300$;
* for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have
$$
g(n_1 + n_2 + \dots + n_{20}) \le f(n_1) + f(n_2) + \dots + f(n_{20}).
$$
Determine the maximum possible value of $g(0) + g(1) + \dots + g(6000)$ over all such pairs of functions.