Question 194
For any integer $n \ge 2$ denote by $A_n$ the set of solutions of the equation
$$
x = \lfloor \frac{x}{2} \rfloor + \lfloor \frac{x}{3} \rfloor + \dots + \lfloor \frac{x}{n} \rfloor.
$$
a) Determine the set $A_2 \cup A_3$.
b) Prove that the set $A = \bigcup_{n \ge 2} A_n$ is finite and find $\max A$.
$$
x = \lfloor \frac{x}{2} \rfloor + \lfloor \frac{x}{3} \rfloor + \dots + \lfloor \frac{x}{n} \rfloor.
$$
a) Determine the set $A_2 \cup A_3$.
b) Prove that the set $A = \bigcup_{n \ge 2} A_n$ is finite and find $\max A$.