# International Mathematics Competition for University Students 2021

Select Year:

IMC 2021
 Information Schedule Problems & Solutions Results Contact

## 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 IMC1994 IMC1995 IMC1996 IMC1997 IMC1998 IMC1999 IMC2000 IMC2001 IMC2002 IMC2003 IMC2004 IMC2005 IMC2006 IMC2007 IMC2008 IMC2009 IMC2010 IMC2011 IMC2012 IMC2013 IMC2014 IMC2015 IMC2016 IMC2017 IMC2018 IMC2019 IMC2020 IMC 2021 