# [Solutions] International Mathematics Competition for University Students 2021

1. Let $A$ be a real $n\times n$ matrix such that $A^3=0$.
a) prove that there is unique real $n\times n$ matrix $X$ that satisfied the equation $$X+AX+XA^2=A$$ b) Express $X$ in terms of $A$.
2. Let $n$ and $k$ be fixed positive integers, and $a$ be arbitrary nonnegative integer. Choose a random $k$-element subset $X$ of $\{1,2,...,k+a\}$ uniformly (i.e., all $k$-element subsets are chosen with the same probability) and, independently of $X$, choose random n-elements subset $Y$ of $\{1,2,..,k+a+n\}$ uniformly. Prove that the probability $$P\left( \text{min}(Y)>\text{max}(X)\right)$$ does not depend on $a$.
3. We say that a positive real number $d$ is $good$ if there exists an infinite squence $a_1,a_2,a_3,...\in (0,d)$ such that for each $n$, the points $a_1,a_2,...,a_n$ partition the interval $[0,d]$ into segments of length at most $\frac{1}{n}$ each. Find $\text{sup}\{d : d \text{ is good}\}$.
4. Let $f:\mathbb{R}\to \mathbb{R}$ be a function. Suppose that for every $\varepsilon >0$ , there exists a function $g:\mathbb{R}\to (0,\infty)$ such that for every pair $(x,y)$ of real numbers, if $|x-y|<\text{min}\{g(x),g(y)\}$, then $|f(x)-f(y)|<\varepsilon$. Prove that $f$ is pointwise limit of a squence of continuous $\mathbb{R}\to \mathbb{R}$ functions i.e., there is a squence $h_1,h_2,...,$ of continuous $\mathbb{R}\to \mathbb{R}$ such that $$\lim_{n\to \infty}h_n(x)=f(x),\,\forall x\in \mathbb{R}.$$
5. Let $A$ be a real $n \times n$ matrix and suppose that for every positive integer $m$ there exists a real symmetric matrix $B$ such that $$2021B = A^m+B^2.$$ Prove that $|\text{det} A| \leq 1$.
6. For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.
7. Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \leq 1\}$. Let $f : D \rightarrow \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that $$|f(0)| \leq \max_{|z|=1} |f(z)p(z)|$$
8. Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?
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