# International Mathematics Competition for University Students 2019

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IMC 2022
 Information Results Problems & Solutions

## IMC2019: Problems on Day 1

Problem 1. Evaluate the product

$\displaystyle \prod_{n=3}^{\infty}\frac{(n^3+3n)^2}{n^6-64} .$

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and

Karen Keryan, Yerevan State University and American University of Armenia, Yerevan

Problem 2. A four-digit number $\displaystyle YEAR$ is called very good if the system

\displaystyle \begin{aligned} Yx+Ey+Az+Rw &= Y \\ Rx + Yy + Ez + Aw &= E \\ Ax + Ry + Yz + Ew &= A \\ Ex + Ay + Rz + Yw &= R \end{aligned}

of linear equations in the variables $\displaystyle x,y,z$ and $\displaystyle w$ has at least two solutions. Find all very good YEARs in the 21st century.

(The 21st century starts in 2001 and ends in 2100.)

Proposed by Tomáš Bárta, Charles University, Prague

Problem 3. Let $\displaystyle f:(-1,1)\to\RR$ be a twice differentiable function such that

$\displaystyle {2f'(x)+xf''(x)\geq1} \quad\text{for $$\displaystyle x\in(-1,1)$}.$$

Prove that

$\displaystyle \int_{-1}^1xf(x)\,\mathrm{d}x\geq\frac{1}{3}.$

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and

Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan

Problem 4. Define the sequence $\displaystyle a_0,a_1,\ldots$ of numbers by the following recurrence:

$\displaystyle a_0=1, \quad a_1=2, \quad (n+3)a_{n+2}=(6n+9)a_{n+1}- na_n \quad \text{for $$\displaystyle n\ge 0$.}$$

Prove that all terms of this sequence are integers.

Proposed by Khakimboy Egamberganov, ICTP, Italy

Problem 5. Determine whether there exist an odd positive integer $\displaystyle n$ and $\displaystyle n\times n$ matrices $\displaystyle A$ and $\displaystyle B$ with integer entries, that satisfy the following conditions:

(1) $\displaystyle \det(B)=1$;

(2) $\displaystyle AB=BA$;

(3) $\displaystyle A^4+4A^2B^2+16B^4=2019I$.

(Here $\displaystyle I$ denotes the $\displaystyle n\times n$ identity matrix.)

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan 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 IMC 2019 IMC2020 IMC2021 IMC2022 