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All-Russian Olympiad
2021 All-Russian Olympiad
7
Nice gcd NT
Nice gcd NT
Source: ARO 2021 11.7
April 20, 2021
number theory
Problem Statement
Find all permutations
(
a
1
,
a
2
,
.
.
.
,
a
2021
)
(a_1, a_2,...,a_{2021})
(
a
1
,
a
2
,
...
,
a
2021
)
of
(
1
,
2
,
.
.
.
,
2021
)
(1,2,...,2021)
(
1
,
2
,
...
,
2021
)
, such that for every two positive integers
m
m
m
and
n
n
n
with difference bigger than
2
0
21
20^{21}
2
0
21
, the following inequality holds:
G
C
D
(
m
+
1
,
n
+
a
1
)
+
G
C
D
(
m
+
2
,
n
+
a
2
)
+
.
.
.
+
G
C
D
(
m
+
2021
,
n
+
a
2021
)
<
2
∣
m
−
n
∣
GCD(m+1, n+a_1)+GCD(m+2, n+a_2)+...+GCD(m+2021, n+a_{2021})<2|m-n|
GC
D
(
m
+
1
,
n
+
a
1
)
+
GC
D
(
m
+
2
,
n
+
a
2
)
+
...
+
GC
D
(
m
+
2021
,
n
+
a
2021
)
<
2∣
m
−
n
∣
.
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