Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer and let
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
be a prime number. Prove that if
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
,
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
,
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
are integers (not necessarily positive) satisfying the equations
then
![a = b = c](/media/m/d/4/b/d4b5f37d2cf6fb668945678e07e47969.png)
.
Proposed by Angelo Di Pasquale, Australia
%V0
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
$$a^n + pb = b^n + pc = c^n + pa$$
then $a = b = c$.
Proposed by Angelo Di Pasquale, Australia