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Ireland National Math Olympiad
1990 Irish Math Olympiad
6
6
Part of
1990 Irish Math Olympiad
Problems
(1)
Equation with 1s and -1s
Source: 1990 IrMO Paper 1 Problem 6
9/30/2017
Let
n
n
n
be a natural number, and suppose that the equation
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+
x
4
x
5
+
⋯
+
x
n
−
1
x
n
+
x
n
x
1
=
0
x_1x_2+x_2x_3+x_3x_4+x_4x_5+\dots +x_{n-1}x_n+x_nx_1=0
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+
x
4
x
5
+
⋯
+
x
n
−
1
x
n
+
x
n
x
1
=
0
has a solution with all the
x
i
x_i
x
i
s equal to
±
1
\pm 1
±
1
. Prove that
n
n
n
is divisible by
4
4
4
.
algebra