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National and Regional Contests
Chile Contests
Chile National Olympiad
1999 Chile National Olympiad
5
x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0 (Chile NMO 1999 P5)
x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0 (Chile NMO 1999 P5)
Source:
November 27, 2021
number theory
multiple
Problem Statement
Consider the numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
that satisfy:
∙
\bullet
∙
x
i
∈
{
−
1
,
1
}
x_i \in \{-1,1\}
x
i
∈
{
−
1
,
1
}
, with
i
=
1
,
2
,
.
.
.
,
n
i = 1, 2,...,n
i
=
1
,
2
,
...
,
n
∙
\bullet
∙
x
1
x
2
x
3
x
4
+
x
2
x
3
x
4
x
5
+
.
.
.
+
x
n
x
1
x
2
x
3
=
0
x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0
x
1
x
2
x
3
x
4
+
x
2
x
3
x
4
x
5
+
...
+
x
n
x
1
x
2
x
3
=
0
Prove that
n
n
n
is a multiple of
4
4
4
.
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