KöMaL 2023 A846

Let $n$ be a positive integer and let vectors $v_1,v_2,\ldots,v_n$ be given in the plain. A flea originally sitting in the origin moves according to the following rule: in the $i^{\text{th}}$ minute (for $i=1,2,\ldots,n$) it will stay where it is with probability $1/2$, moves with vector $v_i$ with probability $1/4$, and moves with vector $-v_i$ with probability $1/4$. Prove that after the $n^{\text{th}}$ minute there exists no point which is occupied by the flea with greater probability than the origin.