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Lusophon Mathematical Olympiad
2021 Lusophon Mathematical Olympiad
4
Cyclic equality
Cyclic equality
Source: Lusophon Mathematical Olympiad 2021 Problem 4
December 19, 2021
algebra
Problem Statement
Let
x
1
,
x
2
,
x
3
,
x
4
,
x
5
∈
R
+
x_1, x_2, x_3, x_4, x_5\in\mathbb{R}^+
x
1
,
x
2
,
x
3
,
x
4
,
x
5
∈
R
+
such that
x
1
2
−
x
1
x
2
+
x
2
2
=
x
2
2
−
x
2
x
3
+
x
3
2
=
x
3
2
−
x
3
x
4
+
x
4
2
=
x
4
2
−
x
4
x
5
+
x
5
2
=
x
5
2
−
x
5
x
1
+
x
1
2
x_1^2-x_1x_2+x_2^2=x_2^2-x_2x_3+x_3^2=x_3^2-x_3x_4+x_4^2=x_4^2-x_4x_5+x_5^2=x_5^2-x_5x_1+x_1^2
x
1
2
−
x
1
x
2
+
x
2
2
=
x
2
2
−
x
2
x
3
+
x
3
2
=
x
3
2
−
x
3
x
4
+
x
4
2
=
x
4
2
−
x
4
x
5
+
x
5
2
=
x
5
2
−
x
5
x
1
+
x
1
2
Prove that
x
1
=
x
2
=
x
3
=
x
4
=
x
5
x_1=x_2=x_3=x_4=x_5
x
1
=
x
2
=
x
3
=
x
4
=
x
5
.
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