# International Mathematics Competition for University Students 2020

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IMC 2020

## 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 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 IMC 2020 