# International Mathematics Competition for University Students 2024

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IMC 2024
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## IMC2024: Day 2, Problem 10

Problem 10. We say that a square-free positive integer $\displaystyle n$ is almost prime if

$\displaystyle n \mid x^{d_1}+x^{d_2}+...+x^{d_k}-kx$

for all integers $\displaystyle x$, where $\displaystyle 1=d_1<d_2<...<d_k=n$ are all the positive divisors of $\displaystyle n$. Suppose that $\displaystyle r$ is a Fermat prime (i.e., it is a prime of the form $\displaystyle 2^{2^m}+1$ for an integer $\displaystyle m\geq 0$), $\displaystyle p$ is a prime divisor of an almost prime integer $\displaystyle n$, and $\displaystyle p\equiv 1\ (\mathrm{mod}\ r)$. Show that, with the above notation, $\displaystyle d_i\equiv 1\ (\mathrm{mod}\ r)$ for all $\displaystyle 1\leq i\leq k$.

(An integer $\displaystyle n$ is called square-free if it is not divisible by $\displaystyle d^2$ for any integer $\displaystyle d>1$.)

Tigran Hakobyan, Yerevan State University, Vanadzor, Armenia

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