Problem

Source: 2021 Taiwan APMO Preliminary First Round

Tags: number theory



Find all positive integers $A,B$ satisfying the following properties: (i) $A$ and $B$ has same digit in decimal. (ii) $2\cdot A\cdot B=\overline{AB}$ (Here $\cdot$ denotes multiplication, $\overline{AB}$ denotes we write $A$ and $B$ in turn. For example, if $A=12,B=34$, then $\overline{AB}=1234$)