International Mathematics Competition for University Students 2017

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IMC 2022
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IMC2017: Problems on Day 1

1. Determine all complex numbers $\lambda$ for which there exist a positive integer $n$ and a real $n\times n$ matrix $A$ such that $A^2=A^T$ and $\lambda$ is an eigenvalue of $A$.

Proposed by: Alexandr Bolbot, Novosibirsk State University

2. Let $f\colon\mathbb{R}\to(0,\infty)$ be a differentiable function, and suppose that there exists a constant $L>0$ such that $$\bigl|f'(x)-f'(y)\bigr| \leq L\bigl|x-y\bigr|$$ for all $x,y$. Prove that $$\big(f'(x)\big)^2 < 2Lf(x)$$ holds for all $x$.

Proposed by: Jan Šustek, University of Ostrava

3. For any positive integer $m$, denote by $P\left(m\right)$ the product of positive divisors of $m$ (e.g. $P(6)=36$). For every positive integer $n$ define the sequence $$a_1(n)=n, \qquad a_{k+1}(n)=P(a_k(n)) \quad (k=1,2,\ldots,2016).$$

Determine whether for every set $S\subseteq\{1,2,\ldots,2017\}$, there exists a positive integer $n$ such that the following condition is satisfied:

For every $k$ with $1\le k\le 2017$, the number $a_k(n)$ is a perfect square if and only if $k\in S$.

Proposed by: Matko Ljulj, University of Zagreb

4. There are $n$ people in a city, and each of them has exactly $1000$ friends (friendship is always symmetric). Prove that it is possible to select a group $S$ of people such that at least $n/2017$ persons in $S$ have exactly two friends in $S$.

Proposed by: Rooholah Majdodin and Fedor Petrov, St. Petersburg State University

5. Let $k$ and $n$ be positive integers with $n\ge k^2-3k+4$, and let $$f(z)=z^{n-1}+c_{n-2}z^{n-2}+\ldots+c_0$$ be a polynomial with complex coefficients such that $$c_0c_{n-2}=c_1c_{n-3}=\ldots=c_{n-2}c_0=0.$$ Prove that $f(z)$ and $z^n-1$ have at most $n-k$ common roots.

Proposed by: Vsevolod Lev and Fedor Petrov, St. Petersburg State University IMC1994 IMC1995 IMC1996 IMC1997 IMC1998 IMC1999 IMC2000 IMC2001 IMC2002 IMC2003 IMC2004 IMC2005 IMC2006 IMC2007 IMC2008 IMC2009 IMC2010 IMC2011 IMC2012 IMC2013 IMC2014 IMC2015 IMC2016 IMC 2017 IMC2018 IMC2019 IMC2020 IMC2021 IMC2022 