In the $100 \times 100$ table in every cell there is natural number. All numbers in same row or column are different. Can be that for every square sum of numbers, that are in angle cells, is square number ?
Problem
Source: St Petersburg Olympiad 2012, Grade 9, P5
Tags: combinatorics, number theory
don2001
04.10.2017 05:39
What are the angle cells
RagvaloD
04.10.2017 13:23
Cells, that are placed in angles of square. For example, if we set for every cell coordinates $(i,j)$ where $i$- column,$j$ -row, then angle cells are $(i,j),(i+k,j),(i,j+k)(i+k,j+k)$ for some $k$
me9hanics
05.10.2017 01:01
Is there a solution? Ohh, I just noticed the SQUARE sum of the numbers must be squares..
don2001
18.10.2017 18:21
Any solution.
don2001
24.03.2018 14:14
After really long time I solved this problem again and here is my solution.
The answer is yes.
Let the $(i,j)$ entry labeled with $a_{i+j-1}$
We filled a table by $a_{k}=c^{2k}$
Hence we have $a_{m}+2a_{m+n}+a_{m+2n}$ is a perfect square for all $m$ and $n$
Here we are done.
SSaad
10.08.2021 19:43
@above how the construction holds? It's still not clear to me what the conditions mean?