Olympiad problems

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Found problems: 2

2023 CIIM, 3

Given a $3 \times 3$ symmetric real matrix $A$, we define $f(A)$ as a $3 \times 3$ matrix with the same eigenvectors of $A$ such that if $A$ has eigenvalues $a$, $b$, $c$, then $f(A)$ has eigenvalues $b+c$, $c+a$, $a+b$ (in that order). We define a sequence of symmetric real $3\times3$ matrices $A_0, A_1, A_2, \ldots$ such that $A_{n+1} = f(A_n)$ for $n \geq 0$. If the matrix $A_0$ has no zero entries, determine the maximum number of indices $j \geq 0$ for which the matrix $A_j$ has any null entries.

2023 Brazil Undergrad MO, 4

Tags: matrix , matrix algebra , linear algebra , eigenvalue and eigenvector

Let $M_2(\mathbb{Z})$ be the set of $2 \times 2$ matrices with integer entries. Let $A \in M_2(\mathbb{Z})$ such that $$A^2+5I=0,$$ where $I \in M_2(\mathbb{Z})$ and $0 \in M_2(\mathbb{Z})$ denote the identity and null matrices, respectively. Prove that there exists an invertible matrix $C \in M_2(\mathbb{Z})$ with $C^{-1} \in M_2(\mathbb{Z})$ such that $$CAC^{-1} = \begin{pmatrix} 1 & 2\\ -3 & -1 \end{pmatrix} \text{ ou } CAC^{-1} = \begin{pmatrix} 0 & 1\\ -5 & 0 \end{pmatrix}.$$