IMC 2003 · Problem 2

Let $a_1, a_2, \dots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the $50$ remaining ones. In this way we get a sequence $b_1, \dots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? (The characteristic of a field is $p$, if $p$ is the smallest positive integer such that $\underbrace{x + x + \dots + x}_{p} = 0$ for any element $x$ of the field. If there exists no such $p$, the characteristic is $0$.)