SIMON-MARAIS 2024 A1

Let $a, b, c$ be real numbers greater than 1 satisfying

$$\lfloor a \rfloor b = \lfloor b \rfloor c = \lfloor c \rfloor a.$$

Prove that $a = b = c$.

(Here, $\lfloor x \rfloor$ denotes the largest integer that is less than or equal to $x$.)