KöMaL 2022 B5269

Let $p \ge 19$ be an odd integer, and color the numbers $0,1,\dots,p-1$ with two colors. For $1\le i\le p$, let $x_i$ denote a random element of the set $\{0,1,\dots,p-1\}$ (the choices are independent, and have uniform distribution). Prove that the probability of the event that $x_1,\dots,x_p$ have the same color and $p$ divides $x_1+\cdots + x_p$ is at least $3/(2^pp)$.