IL-GILLIS 2017 Problem 6

Let $f:\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$ be a function satisfying: [list] [] For any $x_1,x_2,y_1,y_2 \in \mathbb Q$ , $$f\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \leq \frac{f(x_1,y_1)+f(x_2,y_2)}{2}.$$ [] $f(0,0) \leq 0$ . [*] For any $x,y \in \mathbb Q$ satisfying $x^2+y^2>100$ , the inequality $f(x,y)>1$ holds.\

Prove that there is some positive rational number $b$ such that for all rationals $x,y$ , $$f(x,y) \ge b\sqrt{x^2+y^2} - \frac{1}{b}.$$