# International Mathematics Competition for University Students 2021

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IMC 2022
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## IMC2021: Problems on Day 1

Problem 1. Let $\displaystyle A$ be a real $\displaystyle n\times n$ matrix such that $\displaystyle A^3=0$.

(a) Prove that there is a unique real $\displaystyle n\times n$ matrix $\displaystyle X$ that satisfies the equation

$\displaystyle X+AX+XA^2=A.$

(b) Express $\displaystyle X$ in terms of $\displaystyle A$.

Bekhzod Kurbonboev, Institute of Mathematics, Tashkent

Problem 2. Let $\displaystyle n$ and $\displaystyle k$ be fixed positive integers, and let $\displaystyle a$ be an arbitrary non-negative integer. Choose a random $\displaystyle k$-element subset $\displaystyle X$ of $\displaystyle \{1,2,\ldots,k+a\}$ uniformly (i.e., all $\displaystyle k$-element subsets are chosen with the same probability) and, independently of $\displaystyle X$, choose a random $\displaystyle n$-element subset $\displaystyle Y$ of $\displaystyle \{1,\ldots,k+n+a\}$ uniformly.

Prove that the probability

$\displaystyle \mathsf{P}\Big(\min(Y)>\max(X)\Big)$

does not depend on $\displaystyle a$.

Fedor Petrov, St. Petersburg State University

Problem 3. We say that a positive real number $\displaystyle d$ is good if there exists an infinite sequence $\displaystyle a_1,a_2,a_3,\ldots \in (0,d)$ such that for each $\displaystyle n$, the points $\displaystyle a_1,\dots,a_n$ partition the interval $\displaystyle [0,d]$ into segments of length at most $\displaystyle 1/n$ each. Find

$\displaystyle \sup\Big\{d\ \big|\ d \text{ is good}\Big\}.$

Problem 4. Let $\displaystyle f:\RR\to\RR$ be a function. Suppose that for every $\displaystyle \varepsilon > 0$, there exists a function $\displaystyle g:\RR\to(0,\infty)$such that for every pair $\displaystyle (x,y)$ of real numbers,

$\displaystyle \text{if} \quad |x-y| < \min\big\{g(x),g(y)\big\}, \quad\text{then}\quad \big|f(x) - f(y)\big| < \varepsilon.$

Prove that $\displaystyle f$ is the pointwise limit of a sequence of continuous $\displaystyle \RR\to\RR$ functions, i.e., there is a sequence $\displaystyle h_1,h_2,\ldots$ of continuous $\displaystyle \RR\to\RR$ functions such that $\displaystyle \lim\limits_{n\to\infty}h_n(x)=f(x)$ for every $\displaystyle x\in\RR$.

Camille Mau, Nanyang Technological University, Singapore

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