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National and Regional Contests
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pOMA and PErA mathematical olympiads
2023 pOMA
4
Equality algebra
Equality algebra
Source: pOMA 2023/4
November 21, 2023
algebra
Problem Statement
Let
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
be positive real numbers such that
x
1
+
1
x
2
=
x
2
+
1
x
3
=
x
3
+
1
x
4
=
⋯
=
x
n
−
1
+
1
x
n
=
x
n
+
1
x
1
.
x_1+\frac{1}{x_2} = x_2+\frac{1}{x_3} = x_3+\frac{1}{x_4} = \dots = x_{n-1}+\frac{1}{x_n} = x_n+\frac{1}{x_1}.
x
1
+
x
2
1
=
x
2
+
x
3
1
=
x
3
+
x
4
1
=
⋯
=
x
n
−
1
+
x
n
1
=
x
n
+
x
1
1
.
Prove that
x
1
=
x
2
=
x
3
=
⋯
=
x
n
x_1=x_2=x_3=\dots=x_n
x
1
=
x
2
=
x
3
=
⋯
=
x
n
.
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