US-EGMO-TST 2023

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)

Let $ABC$ be a triangle with $AB+AC=3BC$ . The $B$ -excircle touches side $AC$ and line $BC$ at $E$ and $D$ , respectively. The $C$ -excircle touches side $AB$ at $F$ . Let lines $CF$ and $DE$ meet at $P$ . Prove that $\angle PBC = 90^{\circ}$ .

Ray Li