US-EGMO-TST 2023 Problem 2

Consider pairs of functions $(f, g)$ from the set of nonnegative integers to itself such that

• $f(0) + f(1) + f(2) + \cdots + f(42) \le 2022$ ;
• for any integers $a \ge b \ge 0$ , we have $g(a+b) \le f(a) + f(b)$ .

Determine the maximum possible value of $g(0) + g(1) + g(2) + \cdots + g(84)$ over all such pairs of functions.

Evan Chen (adapting from TST3, by Sean Li)