IMC 2019

Evaluate the product $$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}.$$ Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan

A four-digit number $YEAR$ is called very good if the system \begin{align*} Yx+Ey+Az+Rw& =Y Rx+Yy+Ez+Aw & = E
Ax+Ry+Yz+Ew & = A Ex+Ay+Rz+Yw &= R \end{align*} of linear equations in the variables $x,y,z$ and $w$ has at least two solutions. Find all very good $YEAR$ s in the 21st century.

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

Proposed by Tomáš Bárta, Charles University, Prague

Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that $$2f'(x)+xf''(x)\geqslant 1 \text{ for } x\in (-1,1).$$ Prove that $$\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.$$ Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan

Let $(n+3)a_{n+2}=(6n+9)a_{n+1}-na_n$ and $a_0=1$ and $a_1=2$ prove that all the terms of the sequence are integers

Determine whether there exist an odd positive integer $n$ and $n\times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions: [list=1] [] $\det (B)=1$ ;[/] [] $AB=BA$ ;[/] [] $A^4+4A^2B^2+16B^4=2019I$ .[/] [/list] (Here $I$ denotes the $n\times n$ identity matrix.)

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

Let $f,g:\mathbb R\to\mathbb R$ be continuous functions such that $g$ is differentiable. Assume that $(f(0)-g'(0))(g'(1)-f(1))>0$ . Show that there exists a point $c\in (0,1)$ such that $f(c)=g'(c)$ .

Proposed by Fereshteh Malek, K. N. Toosi University of Technology

Let $C=\{4,6,8,9,10,\ldots\}$ be the set of composite positive integers. For each $n\in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$ . Determine whether the following series converges: $$\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.$$ Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan

Let $x_1,\ldots,x_n$ be real numbers. For any set $I\subset\{1,2,…,n\}$ let $s(I)=\sum_{i\in I}x_i$ . Assume that the function $I\to s(I)$ takes on at least $1.8^n$ values where $I$ runs over all $2^n$ subsets of $\{1,2,…,n\}$ . Prove that the number of sets $I\subset \{1,2,…,n\}$ for which $s(I)=2019$ does not exceed $1.7^n$ .

Proposed by Fedor Part and Fedor Petrov, St. Petersburg State University

Determine all positive integers $n$ for which there exist $n\times n$ real invertible matrices $A$ and $B$ that satisfy $AB-BA=B^2A$ .

Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan

$2019$ points are chosen at random, independently, and distributed uniformly in the unit disc $\{(x,y)\in\mathbb R^2: x^2+y^2\le 1\}$ . Let $C$ be the convex hull of the chosen points. Which probability is larger: that $C$ is a polygon with three vertices, or a polygon with four vertices?

Proposed by Fedor Petrov, St. Petersburg State University