Mathematical and Physical Journal
for High Schools
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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.


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

Problems in Mathematics of KöMaL, May 2019