This is a collection of problems from an old notebook from my high school years. I used to collect problems I liked.

  1. Using Viete relations, if \(z_1, ..., z_n\) are roots of a polinomial of the form \(X^n + a_{n-1}X^{n-1} + ... + a_0\) with integral coefficients, then so are \(z_1^k, ..., z_n^k\), for \(k \in \mathbb{N}\). Furthermode, also by Viete relations, the coefficients \(a_{n-1}, ... a_0\) have to be uniformly bounded. Each sequence \(z_i, z_i^2, ...\) must be then periodic, and, therefore, a certain power of \(z_i\) must equal \(1\), for each \(i\).

  2. See https://mathoverflow.net/questions/16721/egz-theorem-erd%C5%91s-ginzburg-ziv

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