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Junior Balkan MO
2018 Junior Balkan MO
3
3
Part of
2018 Junior Balkan MO
Problems
(1)
JBMO 2018, P3
Source: JBMO 2018
6/21/2018
Let
k
>
1
k>1
k
>
1
be a positive integer and
n
>
2018
n>2018
n
>
2018
an odd positive integer. The non-zero rational numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
are not all equal and:
x
1
+
k
x
2
=
x
2
+
k
x
3
=
x
3
+
k
x
4
=
…
=
x
n
−
1
+
k
x
n
=
x
n
+
k
x
1
x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}
x
1
+
x
2
k
=
x
2
+
x
3
k
=
x
3
+
x
4
k
=
…
=
x
n
−
1
+
x
n
k
=
x
n
+
x
1
k
Find the minimum value of
k
k
k
, such that the above relations hold.