OMM 2017 Problem 6

Let $n \geq 2$ and $m$ be positive integers. $m$ ballot boxes are placed in a line. Two players $A$ and $B$ play by turns, beginning with $A$ , in the following manner. Each turn, $A$ chooses two boxes and places a ballot in each of them. Afterwards, $B$ chooses one of the boxes, and removes every ballot from it. $A$ wins if after some turn of $B$ , there exists a box containing $n$ ballots. For each $n$ , find the minimum value of $m$ such that $A$ can guarantee a win independently of how $B$ plays.