# International Mathematics Competition for University Students 2016

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IMC 2019
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## IMC2016: Problems on Day 2

6. Let $(x_1,x_2,\ldots)$ be a sequence of positive real numbers satisfying $\sum_{n=1}^{\infty}\frac{x_n}{2n-1}=1$. Prove that $$\displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{k} \frac{x_n}{k^2} \le2.$$

Proposed by Gerhard J. Woeginger, The Netherlands

7. Today, Ivan the Confessor prefers continuous functions $f:[0,1]\to\mathbb{R}$ satisfying $f(x)+f(y)\geq |x-y|$ for all pairs $x,y\in [0,1]$. Find the minimum of $\int_0^1 f$ over all preferred functions.

Proposed by Fedor Petrov, St. Petersburg State University

8. Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$. Suppose that there exists a function $f:\mathbb{Z}_n\to\mathbb{Z}_n$ satisfying the following three properties:

(i) $f(x)\neq x$,

(ii) $f(f(x))=x$,

(iii) $f(f(f(x+1)+1)+1)=x$ for all $x\in\mathbb{Z}_n$.

Prove that $n\equiv 2 \pmod4$.

Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany

9. Let $k$ be a positive integer. For each nonnegative integer $n$, let $f(n)$ be the number of solutions $(x_1,\ldots,x_k)\in\mathbb{Z}^k$ of the inequality $|x_1|+...+|x_k|\leq n$. Prove that for every $n\ge1$, we have $f(n-1)f(n+1)\leq f(n)^2$.

Proposed by Esteban Arreaga, Renan Finder and José Madrid, IMPA, Rio de Janeiro

10. Let $A$ be a $n\times n$ complex matrix whose eigenvalues have absolute value at most $1$. Prove that $$\|A^n\|\le \dfrac{n}{\ln 2} \|A\|^{n-1}.$$ (Here $\|B\|=\sup\limits_{\|x\|\leq 1} \|Bx\|$ for every $n\times n$ matrix $B$ and $\|x\|=\sqrt{\sum\limits_{i=1}^n |x_i|^2}$ for every complex vector $x\in\mathbb{C}^n$.)

Proposed by Ian Morris 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 IMC 2016 IMC2017 IMC2018 IMC2019 © IMC