| |||||||||||||||

## IMC2024: Problems on Day 2
Mehdi Golafshan & Markus A. Whiteland, University of Liège
\(\displaystyle (A^2 + B^2)(A^4 + B^4) = A^5 + B^5. \) Find all possible values of \(\displaystyle \det(AB)\) for the given \(\displaystyle n\). Sergey Bondarev, Sergey Chernov, Belarusian State University, Minsk
\(\displaystyle x_{n+2}=3x_{n+1}-2x_n+\frac{2^ n}{x_n}\quad\text{for }n\geq 1.\) Prove that \(\displaystyle \displaystyle\lim\limits_{n \to \infty}\frac{x_n}{2^ n}\) exists and satisfies \(\displaystyle \frac{1+\sqrt{3}}{2}\leq\lim\limits_{n \to \infty}\dfrac{x_n}{2^ n}\leq\frac{3}{2}.\) Karen Keryan, Yerevan State University & American University of Armenia, Armenia
(i) the set of all entries of \(\displaystyle A\) is \(\displaystyle \{1,2,\ldots,2t\}\) for some integer \(\displaystyle t\); (ii) the entries are non-decreasing in every row and in every column: \(\displaystyle a_{i,j}\leq a_{i,j+1}\) and \(\displaystyle a_{i,j}\leq a_{i+1,j}\); (iii) equal entries can appear only in the same row or the same column: if \(\displaystyle a_{i,j}=a_{k,\ell}\), then either \(\displaystyle i=k\) or \(\displaystyle j=\ell\); (iv) for each \(\displaystyle s=1,2,\ldots,2t-1\), there exist \(\displaystyle i\ne k\) and \(\displaystyle j\ne \ell\) such that \(\displaystyle a_{i,j}=s\) and \(\displaystyle a_{k,\ell}=s+1\). Prove that for any positive integers \(\displaystyle m\) and \(\displaystyle n\), the number of nice \(\displaystyle m\times n\) matrices is even. For example, the only two nice \(\displaystyle 2\times 3\) matrices are \(\displaystyle \begin{pmatrix} 1&1&1\\2&2&2 \end{pmatrix}\) and \(\displaystyle \begin{pmatrix} 1&1&3\\2&4&4 \end{pmatrix}\). Fedor Petrov, St Petersburg State University
\(\displaystyle n \mid x^{d_1}+x^{d_2}+...+x^{d_k}-kx \) for all integers \(\displaystyle x\), where \(\displaystyle 1=d_1<d_2<...<d_k=n\) are all the positive divisors of \(\displaystyle n\). Suppose that \(\displaystyle r\) is a Fermat prime (i.e., it is a prime of the form \(\displaystyle 2^{2^m}+1\) for an integer \(\displaystyle m\geq 0\)), \(\displaystyle p\) is a prime divisor of an almost prime integer \(\displaystyle n\), and \(\displaystyle p\equiv 1\ (\mathrm{mod}\ r)\). Show that, with the above notation, \(\displaystyle d_i\equiv 1\ (\mathrm{mod}\ r)\) for all \(\displaystyle 1\leq i\leq k\). (An integer \(\displaystyle n\) is called Tigran Hakobyan, Yerevan State University, Vanadzor, Armenia | |||||||||||||||

© IMC |