Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 752. (May 2019)

A. 752. Let $\displaystyle k$, $\displaystyle s$ and $\displaystyle n$ be positive integers such that $\displaystyle s<(2k+1)^2$, and let $\displaystyle R$ be the set of lattice points $\displaystyle (x,y)$ in the plane, satisfying $\displaystyle 1\le x,y\le n$. On the grid $\displaystyle R$ we perform the following procedure. Initially, we colour one point of $\displaystyle R$ green; all other points in $\displaystyle R$ are coloured white. On every move, we choose a square $\displaystyle S$, consisting of $\displaystyle {\color{red}(2k+1)\times(2k+1)}$ lattice points in such a way that the center of $\displaystyle S$ is green and it contains at least $\displaystyle s$ white points; then we re-colour $\displaystyle s$ white points of $\displaystyle S$ to green. We repeat this step as long as there is a suitable square $\displaystyle S$.

We say that the number $\displaystyle s$ is $\displaystyle k$-sparse, if there exists a positive real number $\displaystyle C$ such that, for every $\displaystyle n$, for every choice of the initial green point, and for every valid sequence of moves, the total number of green points in the grid cannot exceed $\displaystyle Cn$.

Determine the least $\displaystyle k$-sparse positive integer $\displaystyle s$ in terms of $\displaystyle k$.

Proposed by: Nikolai Beluhov (Stara Zagora, Bulgaria)

(7 pont)

Deadline expired on June 11, 2019.

### Statistics:

 3 students sent a solution. 7 points: Schrettner Jakab. 4 points: 1 student. 0 point: 1 student.

Problems in Mathematics of KöMaL, May 2019