# International Mathematics Competition for University Students 2020

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IMC 2022
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## IMC2020: Problems on Day 1

Problem 1. Let $\displaystyle n$ be a positive integer. Compute the number of words $\displaystyle w$ (finite sequences of letters) that satisfy all the following three properties:

(1) $\displaystyle w$ consists of $\displaystyle n$ letters, all of them are from the alphabet $\displaystyle \{\texttt{a},\texttt{b},\texttt{c},\texttt{d}\}$;

(2) $\displaystyle w$ contains an even number of letters $\displaystyle \texttt{a}$;

(3) $\displaystyle w$ contains an even number of letters $\displaystyle \texttt{b}$.

(For example, for $\displaystyle n=2$ there are $\displaystyle 6$ such words: $\displaystyle \texttt{aa}$, $\displaystyle \texttt{bb}$, $\displaystyle \texttt{cc}$, $\displaystyle \texttt{dd}$, $\displaystyle \texttt{cd}$ and $\displaystyle \texttt{dc}$.)

Armend Sh. Shabani, University of Prishtina

Problem 2. Let $\displaystyle A$ and $\displaystyle B$ be $\displaystyle n\times n$ real matrices such that

$\displaystyle \textrm{rk}(AB-BA+I)=1$

where $\displaystyle I$ is the $\displaystyle n\times n$ identity matrix.

Prove that

$\displaystyle \tr(ABAB)-\tr(A^2B^2)=\frac12 n(n-1).$

($\displaystyle \textrm{rk}(M)$ denotes the rank of matrix $\displaystyle M$, i.e., the maximum number of linearly independent columns in $\displaystyle M$. $\displaystyle \tr(M)$ denotes the trace of $\displaystyle M$, that is the sum of diagonal elements in $\displaystyle M$.)

Rustam Turdibaev, V. I. Romanovskiy Institute of Mathematics

Problem 3. Let $\displaystyle d\ge 2$ be an integer. Prove that there exists a constant $\displaystyle C(d)$ such that the following holds: For any convex polytope $\displaystyle K\subset \mathbb{R}^d$, which is symmetric about the origin, and any $\displaystyle \varepsilon\in (0,1)$, there exists a convex polytope $\displaystyle L\subset \mathbb R^d$ with at most $\displaystyle C(d)\varepsilon^{1-d}$ vertices such that

$\displaystyle (1-\varepsilon)K\subseteq L \subseteq K.$

(For a real $\displaystyle \alpha$, a set $\displaystyle T\subset \mathbb{R}^d$ with nonempty interior is a convex polytope with at most $\displaystyle \alpha$ vertices, if $\displaystyle T$ is a convex hull of a set $\displaystyle X\subset\mathbb R^d$ of at most $\displaystyle \alpha$ points, i.e., $\displaystyle T=\{\sum_{x\in X} t_xx\ |\ t_x\ge 0, \sum_{x\in X} t_x = 1\}$. For a real $\displaystyle \lambda$, put $\displaystyle \lambda K=\{\lambda x\ |\ x\in K\}$. A set $\displaystyle T\subset \mathbb{R}^d$ is symmetric about the origin if $\displaystyle (-1) T = T$.)

Fedor Petrov, St. Petersburg State University

Problem 4. A polynomial $\displaystyle p$ with real coefficients satisfies the equation $\displaystyle p(x+1)-p(x)=x^{100}$ for all $\displaystyle x\in\mathbb{R}$. Prove that $\displaystyle p(1-t)\geqslant p(t)$ for $\displaystyle 0\leqslant t\leqslant 1/2$.

Daniil Klyuev, St. Petersburg State University

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