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## IMC2020: Problems on Day 1
(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
\(\displaystyle \textrm{rk}(AB-BA+I)=1 \) 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
\(\displaystyle (1-\varepsilon)K\subseteq L \subseteq K. \) (For a real \(\displaystyle \alpha\), a set \(\displaystyle T\subset \mathbb{R}^d\) with nonempty interior is a Fedor Petrov, St. Petersburg State University
Daniil Klyuev, St. Petersburg State University | |||||||||||||

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