I've been doing some polynomial excersises lately and in that one I got completly stuck.
Let $m \geqslant 1$ be natural number and $P(x)$ polynomial with integer coefficients which has at least three different integer roots. Prove that $P(x)+5^m$ has no more than one integer root.
At first I considered the easiest case: $(x-x_1)(x-x_2)(x-x_3)+5$, but it did not turned out in anything helpful, so I am seeking for some clues on how to crack that problem.
Also, I'd like to ask for as elementary hint/solution as possible since this question is from (inactive) high school contest. https://om.mimuw.edu.pl/static/app_main/problems/om48_1.pdf