5

Part of 2017 Czech-Polish-Slovak Junior Match

Problems(2)

real numbers from [0, 1] into (mn + 1) x (mn + 1) table

Source: Czech-Polish-Slovak Junior Match 2017, individual p5 CPSJ

Each field of the table

(mn + 1) \times (mn + 1)

contains a real number from the interval

[0, 1]

. The sum the numbers in each square section of the table with dimensions

n x n

n

. Determine how big it can be sum of all numbers in the table.

1,2,3 in 100x100 square table , balanced 2x2 subtables

Source: Czech-Polish-Slovak Junior Match 2017, Team p5 CPSJ

In each square of the

100\times 100

square table, type

1, 2

3

. Consider all subtables

m \times n

m = 2

n = 2

. A subtable will be called balanced if it has in its corner boxes of four identical numbers boxes . For as large a number

k

prove, that we can always find

k

balanced subtables, of which no two overlap, i.e. do not have a common box.

combinatoricstablenumbers in a table