Arranging numbers in a circle

Here's a question I found from a Chinese book, (which contains no answers), and I found it to be very interesting.

Suppose we have $\displaystyle n$ numbers whose sum is 0.

Arrange them in a fixed order $\displaystyle (a_1,a_2,a_3,a_4...a_n)$ in a circular manner.

Show that there exists an integer m, such that $\displaystyle a_m, 1 \geq m \geq n$, and the sum of any number (1 to n) of consecutive terms starting with $\displaystyle a_m$ is non-negative. (Clockwise direction)

It seems that it is true, after trying with a few numbers.

How do you actually prove it?

Thanks in advance.