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Found problems: 2

1997 Miklós Schweitzer, 4
Tags: linear algebra , binary matrix
An elementary change in a 0-1 matrix is ​​a change in an element and with it all its horizontal, vertical, and diagonal neighbors (0 to 1 or 1 to 0). Can any 1791 x 1791 0-1 matrix be transformed into a zero matrix with elementary changes?

2021 Alibaba Global Math Competition, 3
Tags: matrices , binary matrix
Given positive integers $k \ge 2$ and $m$ sufficiently large. Let $\mathcal{F}_m$ be the infinite family of all the (not necessarily square) binary matrices which contain exactly $m$ 1's. Denote by $f(m)$ the maximum integer $L$ such that for every matrix $A \in \mathcal{F}_m$, there always exists a binary matrix $B$ of the same dimension such that (1) $B$ has at least $L$ 1-entries; (2) every entry of $B$ is less or equal to the corresponding entry of $A$; (3) $B$ does not contain any $k \times k$ all-1 submatrix. Show the equality \[\lim_{m \to \infty} \frac{\ln f(m)}{\ln m}=\frac{k}{k+1}.\]