I saw something very simillar to this in the winter camp, so I think it's not that good.

Actually it's 99% same problem. https://artofproblemsolving.com/community/q8h1321555p7119978 걍 똑같이 냈네 ㅋㅋㅋㅋ

Can u post the other problems as well

Hypernova wrote: I saw something very simillar to this in the winter camp, so I think it's not that good. I agree. These types of problems are not good for competitions since it takes only <5 minutes for who had ever seen the problem, but more than an hour for who didn't(especially for this problem, since the $p=7k-1$ version of this problem appears on every single winter camp's exercises). The substitution $m \leftarrow x+\frac{1}{x}$ is not easy to think..

A similar, but harder problem appeared on Brazilian MO 2017 (https://artofproblemsolving.com/community/c6h1556461p9495218), But the intuition is to try to make the $\Phi_7(X)$ appear somehow, instead of the polynomial with $m$ and degree 3. As we want to double the degree in order for the cyclotomic polynomial to appear, some substitution like $m \rightarrow x+\frac{1}{x}$ must be a good idea, and in fact it works. Now it is enough to show that there is some $a$ with order 7 mod $p$, which is a well-known fact that can be easily proven with cyclotomic polynomials. $\blacksquare$

So late.. Oops I think they wanted this solution

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