BALTIC-WAY 2003 Problem 6

Let $n \geq 2$ and $d \geq 1$ be integers with $d \mid n$, and let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers such that $x_{1}+x_{2}+\cdots+x_{n}=0$. Prove that there are at least $\left(\begin{array}{l}n-1 \\ d-1\end{array}\right)$ choices of $d$ indices $1 \leq i_{1}<i_{2}<\cdots<i_{d} \leq n$ such that $x_{i_{1}}+x_{i_{2}}+\cdots+x_{i_{d}} \geq 0$.