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National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2021 Bosnia and Herzegovina Junior BMO TST
4
4
Part of
2021 Bosnia and Herzegovina Junior BMO TST
Problems
(1)
beautiful 3xn board with integers from 1 to n
Source: 2021 JBMO TST Bosnia and Herzegovina P4
10/7/2022
Let
n
n
n
be a nonzero natural number and let
S
=
{
1
,
2
,
.
.
.
,
n
}
S = \{1, 2, . . . , n\}
S
=
{
1
,
2
,
...
,
n
}
. A
3
×
n
3 \times n
3
×
n
board is called beautiful if it can be completed with numbers from the set
S
S
S
like this as long as the following conditions are met:
∙
\bullet
∙
on each line, each number from the set S appears exactly once,
∙
\bullet
∙
on each column the sum of the products of two numbers on that column is divisible by
n
n
n
(that is, if the numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are written on a column, it must be
a
b
+
b
c
+
c
a
ab + bc + ca
ab
+
b
c
+
c
a
be divisible by
n
n
n
). For which values of the natural number
n
n
n
are there beautiful tables ¸and for which values do not exist? Justify your answer.
combinatorics