Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any $n*n$ square is less or equal than $1$. Prove that the absolute value of the sum of the numbers from any $m*n$ rectangular is less or equal than $4$.
Problem
Source: Danube Mathematical Competition 2017, Romania
Tags: combinatorics, abstract algebra, absolute value
CZRorz
30.10.2017 07:46
Any solution?
me9hanics
30.10.2017 23:25
What if we take $m>4n$ and fill each square with $\frac{1}{n^2}$?
GGPiku
30.10.2017 23:29
misinnyo wrote: What if we take $m>4n$ and fill each square with $\frac{1}{n^2}$? Surely you’ll find a square with the sum over 1, which is what the problem asked not to happen.
programjames1
31.10.2017 01:18
Actually, I don't think so. the largest sized square will be a $n$ by $n$ sized square, and will have a sum of $1$. Every other square will have less than that.
GGPiku
31.10.2017 10:05
@above keep in mind that the conditions have to be true for any square. Anyway, here is the official solution:
Attachments:
MDC2017_seniori-solutii.pdf (390kb)
me9hanics
18.11.2017 21:37
Well then the exercise was translated wrongly. It should be edited