It is from Tuymaada 2024. Are you sure that we can discuss problems now? As far I know, they weren't posted anywhere in open resource.

NO_SQUARES wrote: It is from Tuymaada 2024. Are you sure that we can discuss problems now? As far I know, they weren't posted anywhere in open resource. If the contest is over, why is it not allowed?

crocodilepradita wrote: Given a board with size $25\times 25$. Some $1\times 1$ squares are marked, so that for each $13\times 13$ and $4\times 4$ sub-boards, there are atleast $\frac{1}{2}$ marked parts of the sub-board. Find the least possible amount of marked squares in the entire board. Answer: $313$. Estimation: Divide square on $18$ squares with side $4$ and $2$ squares with side $13$ (as on picture). It is not hard to see that there are should be al least $313$ marked cells. (In squares $12 \times 12$ there are at least $144/2$ and in squares $13 \times 13$ there are at least $170/2$ marked cells). Example: Mark the cells as in a chess coloring book in which the center is not colored, and also mark the center. The picture shows an example for a square with side $5$. Chess coloring gives $\frac{25^2-1}{2}$ and center gives $1$ marked cell, so there are $\frac{25^2-1}{2}+1=313$ marked cells. The chess coloring allows you to achieve what you want for squares with side $4$, and for squares with side $13$, a maximum of $1$ marked cell is missing. It remains to be noted that all such squares contain the center of the square with side $25$, which is also marked.

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Answer: $313$ Construction: Label the grid $G=\{(x,y)\;|\;0\leq x,y\leq 24, x,y\in \mathbb{Z}\}$. Mark the square in a checkerboard pattern marking the squares $(x,y)$ if $x+y$ is odd or if $(x,y)=(12,12)$. Every $4\times 4$ sub-grid clearly works and every $13\times 13$ sub-grid contains at least $84$ marked squares on the checkerboard and must also contain the central square. Bound: Consider the $13\times 13$ sub-grids $[0,12]\times [0,12]$ and $[12,24]\times[12,24]$. The $12\times 12$ sub-grids $[0,11]\times[13,24]$ and $[13,24]\times [0,11]$ can each be divided into $9$ smaller $4\times 4$ sub-grids. If the central square is unmarked then we need at least $85$ marked squares in both $13\times 13$ grids and $8$ marked squares in all $18$ of the $4\times 4$ grids, resulting in $314$ marked squares. If the central square is marked then we need at least $84$ additional marked squares in both $13\times 13$ grids and $8$ marked squares in all $18$ of the $4\times 4$ grids, resulting in $313$ marked squares.