Question 69
For a real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$. Consider the function
$$
f(x, y)=\sqrt{M(M+1)}(|x-m|+|y-m|)
$$
where $M=\max (\lfloor x\rfloor,\lfloor y\rfloor)$ and $m=\min (\lfloor x\rfloor,\lfloor y\rfloor)$. The set of all real numbers $(x, y)$ such that $2 \leq x, y \leq 2022$ and $f(x, y) \leq 2$ can be expressed as a finite union of disjoint regions in the plane. The sum of the areas of these regions can be expressed as a fraction $a / b$ in lowest terms. What is the value of $a+b$ ?
$$
f(x, y)=\sqrt{M(M+1)}(|x-m|+|y-m|)
$$
where $M=\max (\lfloor x\rfloor,\lfloor y\rfloor)$ and $m=\min (\lfloor x\rfloor,\lfloor y\rfloor)$. The set of all real numbers $(x, y)$ such that $2 \leq x, y \leq 2022$ and $f(x, y) \leq 2$ can be expressed as a finite union of disjoint regions in the plane. The sum of the areas of these regions can be expressed as a fraction $a / b$ in lowest terms. What is the value of $a+b$ ?