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KoMaL A Problems
KoMaL A Problems 2021/2022
A. 826
A. 826
Part of
KoMaL A Problems 2021/2022
Problems
(1)
Numbers joined by an antelope
Source: KoMaL A. 826
5/11/2022
An antelope is a chess piece which moves similarly to the knight: two cells
(
x
1
,
y
1
)
(x_1,y_1)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
(x_2,y_2)
(
x
2
,
y
2
)
are joined by an antelope move if and only if
{
∣
x
1
−
x
2
∣
,
∣
y
1
−
y
2
∣
}
=
{
3
,
4
}
.
\{|x_1-x_2|,|y_1-y_2|\}=\{3,4\}.
{
∣
x
1
−
x
2
∣
,
∣
y
1
−
y
2
∣
}
=
{
3
,
4
}
.
The numbers from
1
1
1
to
1
0
12
10^{12}
1
0
12
are placed in the cells of a
1
0
6
×
1
0
6
10^6\times 10^6
1
0
6
×
1
0
6
grid. Let
D
D
D
be the set of all absolute differences of the form
∣
a
−
b
∣
|a-b|
∣
a
−
b
∣
, where
a
a
a
and
b
b
b
are joined by an antelope move in the arrangement. How many arrangements are there such that
D
D
D
contains exactly four elements?Proposed by Nikolai Beluhov, Bulgaria
combinatorics
komal