OSN 2019 Problem 5

Given that $a$ and $b$ are real numbers such that for infinitely many positive integers $m$ and $n$ , $$\lfloor an + b \rfloor \ge \lfloor a + bn \rfloor$$ $$\lfloor a + bm \rfloor \ge \lfloor am + b \rfloor$$ Prove that $a = b$ .