[Solutions] International Mathematics Competition for University Students 2019

1. Evaluate the product $$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}.$$
2. 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 $2100$.)
3. Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that $$2f’(x)+xf''(x)\geqslant 1 \quad \text{ for } x\in (-1,1).$$ Prove that $$\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.$$
4. 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.
5. 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: $\det (B)=1$; $AB=BA$;  and $$A^4+4A^2B^2+16B^4=2019I.$$ Here $I$ denotes the $n\times n$ identity matrix.
6. 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 $f(c)=g'(c)$.
7. 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 $$\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.$$
8. 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 $1.7^n$.
9. Determine all positive integers $n$ for which there exist $n\times n$ real invertible matrices $A$ and $B$ that satisfy $$AB-BA=B^2A.$$
10. $2019$ 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?