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National and Regional Contests
Poland Contests
Polish MO Finals
1990 Polish MO Finals
2
cyclic inequality with fractions
cyclic inequality with fractions
Source: Problem 2, Polish NO 1990
September 30, 2005
inequalities
inequalities unsolved
Problem Statement
Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, . . . , x_n
x
1
,
x
2
,
...
,
x
n
be positive numbers. Prove that
∑
i
=
1
n
x
i
2
x
i
2
+
x
i
+
1
x
i
+
2
≤
n
−
1
\sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1
i
=
1
∑
n
x
i
2
+
x
i
+
1
x
i
+
2
x
i
2
≤
n
−
1
Where
x
n
+
1
=
x
1
x_{n+1}=x_1
x
n
+
1
=
x
1
and
x
n
+
2
=
x
2
x_{n+2}=x_2
x
n
+
2
=
x
2
.
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