[Solutions] Vojtěch Jarník International Mathematical Competition 2019

Category I

1. Let $\{a_n \}_{n=0}^{\infty}$ be a sequence given recrusively such that $$a_0=1,\quad a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2},\,\forall n\geq 0.$$ Show that a) $a_n$ is a positive integer.
b) $a_n a_{n+1}-1$ is a square of an integer.
2. A triplet of polynomials $u,v,w \in \mathbb{R}[x,y,z]$ is called smart if there exists polynomials $P,Q,R\in \mathbb{R}[x,y,z]$ such that the following polynomial identity holds $$u^{2019}P +v^{2019 }Q+w^{2019} R=2019.$$ a) Is the triplet of polynomials $u=x+2y+3$, $v=y+z+2$, $w=x+y+z$ smart?.
b) Is the triplet of polynomials $u=x+2y+3$, $v=y+z+2$, $w=x+y-z$ smart?.
3. For an invertible $n\times n$ matrix $M$ with integer entries we define a sequence $\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}$ by the recurrence $M_0=M$, $M_{i+1}=(M_i^T)^{-1}M_i$ for $i\geq 0$. Find the smallest integer $n\geq 2$ for wich there exists a normal $n\times n$ matrix with integer entries such that its sequence $\mathcal{S}_M$ is not constant and has period $P=7$ i.e $M_{i+7}=M_i$. ($M^T$ means the transpose of a matrix $M$. A square matrix is called normal if $M^T M=M M^T$ holds.)
4. Determine the largest constant $K\geq 0$ such that $$\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2$$holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$.

Category II

1. a) Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the conditions $$x*x*y=y \;\;\; \text{and}\; \;\; y*x*x=y$$for every $x,y\in A$ imply commutativity of $*$?.
b) Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the condition $$x*x*y=y$$for every $x,y\in A$ implies commutativity of $*$?
2. Find all twice differentiable functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f''(x) \cos(f(x))\geq(f'(x))^2 \sin(f(x))$$ for every $x\in \mathbb{R}$.
3. Let $p$ be an even non-negative continous function with $\int _{\mathbb{R}} p(x) dx =1$ and let $n$ be a positive integer. Let $\xi_1,\xi_2,\xi_3 \dots ,\xi_n$ be independent identically distributed random variables with density function $p$. Define \begin{align*} X_{0} & = 0 \\ X_{1} & = X_0+ \xi_1 \\ & \vdots \\ X_{n} & = X_{n-1} + \xi_n \end{align*} Prove that the probability that all random variables $X_1,X_2 \dots X_{n-1}$ lie between $X_0$ and $X_n$ is $\frac{1}{n}$.
4. Let $D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \}$. Let $n \geq 1$ and let $a_1,a_2,\dots a_n \in D$ be distinct complex numbers. Define $$f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}}.$$ Prove that $f'$ has at least one root in $D$.