IMC 2024 Problem 10

We say that a square-free positive integer $n$ is almost prime if $$n \mid x^{d_1}+x^{d_2}+\dots+x^{d_k}-kx$$ for all integers $x$ , where $1=d_1<d_2<\dots<d_k=n$ are all the positive divisors of $n$ . Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m}+1$ for an integer $m \ge 0$ ), $p$ is a prime divisor of an almost prime integer $n$ , and $p \equiv 1 \pmod{r}$ . Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$ . (An integer $n$ is called square-free if it is not divisible by $d^2$ for any integer $d>1$ .)