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All-Russian Olympiad
2021 All-Russian Olympiad
2
system of equations to be satisfied
system of equations to be satisfied
Source: All-Russian 2021/10.2
April 19, 2021
number theory
Problem Statement
Find all sets of positive integers
{
x
1
,
x
2
,
…
,
x
20
}
\{x_1, x_2, \dots, x_{20}\}
{
x
1
,
x
2
,
…
,
x
20
}
such that
x
i
+
2
2
=
l
c
m
(
x
i
+
1
,
x
i
)
+
l
c
m
(
x
i
,
x
i
−
1
)
x_{i+2}^2=lcm(x_{i+1}, x_{i})+lcm(x_{i}, x_{i-1})
x
i
+
2
2
=
l
c
m
(
x
i
+
1
,
x
i
)
+
l
c
m
(
x
i
,
x
i
−
1
)
for
i
=
1
,
2
,
…
,
20
i=1, 2, \dots, 20
i
=
1
,
2
,
…
,
20
where
x
0
=
x
20
,
x
21
=
x
1
,
x
22
=
x
2
x_0=x_{20}, x_{21}=x_1, x_{22}=x_2
x
0
=
x
20
,
x
21
=
x
1
,
x
22
=
x
2
.
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