Question 91
Define $M = \{x \in \mathbb{R} \mid \lfloor nx \rfloor + \lfloor (n+1)x \rfloor = \lfloor (2n+1)x \rfloor, \forall n \in \mathbb{N}\}$.
a) Prove that, if $x \in M \cap [\frac{1}{2}, 1)$, then $\lfloor nx \rfloor = \lfloor \frac{n}{2} \rfloor, \forall n \in \mathbb{N}$.
b) Find the cardinal of the set $M \cap [0, 2023]$.
a) Prove that, if $x \in M \cap [\frac{1}{2}, 1)$, then $\lfloor nx \rfloor = \lfloor \frac{n}{2} \rfloor, \forall n \in \mathbb{N}$.
b) Find the cardinal of the set $M \cap [0, 2023]$.