# International Mathematics Competition for University Students 2021

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IMC 2022
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## IMC2021: Problems on Day 2

Problem 5. Let $\displaystyle A$ be a real $\displaystyle n\times n$ matrix and suppose that for every positive integer $\displaystyle m$ there exists a real symmetric matrix $\displaystyle B$ such that

$\displaystyle 2021B=A^m+B^2.$

Prove that $\displaystyle |\det{A}|\le 1$.

Rafael Filipe dos Santos, Instituto Militar de Engenharia, Rio de Janeiro

Problem 6. For a prime number $\displaystyle p$, let $\displaystyle \mathrm{GL}_2(\ZZ/p\ZZ)$ be the group of invertible $\displaystyle 2 \times 2$ matrices of residues modulo $\displaystyle p$, and let $\displaystyle S_p$ be the symmetric group (the group of all permutations) on $\displaystyle p$ elements. Show that there is no injective group homomorphism $\displaystyle \varphi : \mathrm{GL}_2(\ZZ/p\ZZ) \to S_p$.

Thiago Landim, Sorbonne University, Paris

Problem 7. Let $\displaystyle D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\displaystyle \{z\ :\ |z|\le 1\}$. Let $\displaystyle f:D \to \mathbb{C}$ be a holomorphic function, and let $\displaystyle p(z)$ be a monic polynomial. Prove that

$\displaystyle \big|f(0)\big| \le \max_{|z|=1} \big|f(z)p(z)\big|.$

Lars Hörmander

Problem 8. Let $\displaystyle n$ be a positive integer. At most how many distinct unit vectors can be selected in $\displaystyle \RR^n$ such that from any three of them, at least two are orthogonal?

Alexander Polyanskii, Moscow Institute of Physics and Technology;
based on results of Paul Erdős and Moshe Rosenfeld

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