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1985 IMO Longlists
40
Prove that 4|n (Old Problem)
Prove that 4|n (Old Problem)
Source:
September 13, 2010
modular arithmetic
algebra proposed
algebra
Problem Statement
Each of the numbers
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \dots, x_n
x
1
,
x
2
,
…
,
x
n
equals
1
1
1
or
−
1
-1
−
1
and
∑
i
=
1
n
x
i
x
i
+
1
x
i
+
2
x
i
+
3
=
0.
\sum_{i=1}^n x_i x_{i+1} x_{i+2} x_{i+3} =0.
i
=
1
∑
n
x
i
x
i
+
1
x
i
+
2
x
i
+
3
=
0.
where
x
n
+
i
=
x
i
x_{n+i}=x_i
x
n
+
i
=
x
i
for all
i
i
i
. Prove that
4
∣
n
4\mid n
4
∣
n
.
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