2008 IMO Shortlist N1

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a, b, c$ are integers (not necessarily positive) satisfying the equations $$a^{n}+p b=b^{n}+p c=c^{n}+p a,$$ then $a=b=c$.

设$n$为正整数，$p$为素数。证明如果 $a, b, c$ 是满足方程 $$a^{n}+p b=b^{n}+p c=c^{n}+p a,$$ 的整数(不一定是正数)，则 $a=b=c$。