Question 21
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 $a=b=c$.
$$
a^{n}+p b=b^{n}+p c=c^{n}+p a,
$$
then $a=b=c$.