RO-NMO 2010 Problem 93

For any integer $n\ge 2$ denote by $A_n$ the set of solutions of the equation $$x=\left\lfloor\frac{x}{2}\right\rfloor+\left\lfloor\frac{x}{3}\right\rfloor+\cdots+\left\lfloor\frac{x}{n}\right\rfloor .$$ 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 $\max A$ .

Dan Nedeianu & Mihai Baluna