2

Part of 2017 Danube Mathematical Olympiad

Problems(2)

Placing Numbers On A Board

Source: Mathematical Danube Competition 2017, Juniors P2

n\geq 3

be a positive integer. Consider an

n\times n

square. In each cell of the square, one of the numbers from the set

M=\{1,2,\ldots,2n-1\}

is to be written. One such filling is called good if, for every index

1\leq i\leq n,

i

i,

together, contain all the elements of

M

.
[*]Prove that there exists

n\geq 3

for which a good filling exists. [*]Prove that for

n=2017

there is no good filling of the

n\times n

combinatoricsromania

Source: Danube Mathematical Competition 2017, Romania

Let n be a positive interger. Let n real numbers be wrote on a paper. We call a "transformation" :choosing 2 numbers

a,b

and replace both of them with

a*b

. Find all n for which after a finite number of transformations and any n real numbers, we can have the same number written n times on the paper.