# International Mathematics Competition for University Students 2024

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IMC 2024
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## IMC2024: Problems on Day 1

Problem 1. Determine all pairs $\displaystyle (a,b) \in \mathbb{C} \times \mathbb{C}$ satisfying

$\displaystyle |a| = |b| = 1 \quad \text{and} \quad a + b + a\overline{b} \in \mathbb{R}.$

Mike Daas, Universiteit Leiden

Problem 2. For $\displaystyle n=1,2,\ldots$ let

$\displaystyle S_n = \log\left( \sqrt[n^2]{1^1\cdot 2^2\cdot\ldots\cdot n^n}\right) - \log (\sqrt{n}),$

where $\displaystyle \log$ denotes the natural logarithm. Find $\displaystyle \mathop{\lim}\limits_{n\to\infty} S_n$.

Sergey Chernov, Belarusian State University, Minsk

Problem 3. For which positive integers $\displaystyle n$ does there exist an $\displaystyle n\times n$ matrix $\displaystyle A$ whose entries are all in $\displaystyle \{0,1\}$, such that $\displaystyle A^2$ is the matrix of all ones?

Alex Avdiushenko, Neapolis University Paphos, Cyprus

Problem 4. Let $\displaystyle g$ and $\displaystyle h$ be two distinct elements of a group $\displaystyle G$, and let $\displaystyle n$ be a positive integer. Consider a sequence $\displaystyle w=(w_1,w_2,\ldots)$ which is not eventually periodic and where each $\displaystyle w_i$ is either $\displaystyle g$ or $\displaystyle h$. Denote by $\displaystyle H$ the subgroup of $\displaystyle G$ generated by all elements of the form $\displaystyle w_{k}w_{k+1}\ldots w_{k+n-1}$ with $\displaystyle k\geq 1$. Prove that $\displaystyle H$ does not depend on the choice of the sequence $\displaystyle w$ (but may depend on $\displaystyle n$).

Ivan Mitrofanov, Saarland University

Problem 5. Let $\displaystyle n>d$ be positive integers. Choose $\displaystyle n$ independent, uniformly distributed random points $\displaystyle x_1,\ldots,x_n$ in the unit ball $\displaystyle B\subset \mathbb{R}^d$ centered at the origin. For a point $\displaystyle p\in B$ denote by $\displaystyle f(p)$ the probability that the convex hull of $\displaystyle x_1,\ldots,x_n$ contains $\displaystyle p$. Prove that if $\displaystyle p,q\in B$ and the distance of $\displaystyle p$ from the origin is smaller than the distance of $\displaystyle q$ from the origin, then $\displaystyle f(p)\geq f(q)$.

Fedor Petrov, St Petersburg State University

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