International Mathematics Competition
for University Students
2021

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

  Hint    Solution  

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

  Hint    Solution  

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

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

Josef Tkadlec

  Hint    Solution  

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

  Hint    Solution