# 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 IMC1994 IMC1995 IMC1996 IMC1997 IMC1998 IMC1999 IMC2000 IMC2001 IMC2002 IMC2003 IMC2004 IMC2005 IMC2006 IMC2007 IMC2008 IMC2009 IMC2010 IMC2011 IMC2012 IMC2013 IMC2014 IMC2015 IMC2016 IMC2017 IMC2018 IMC2019 IMC2020 IMC 2021 IMC2022 