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National and Regional Contests
Cuba Contests
Cuba MO
2005 Cuba MO
9
sum x_k/(x_ky_k + x_{k+1}) >1/2n
sum x_k/(x_ky_k + x_{k+1}) >1/2n
Source: 2005 Cuba MO 2.9
September 15, 2024
algebra
inequalities
Problem Statement
Let
x
1
,
x
2
,
…
,
x
n
x_1, x_2, …, x_n
x
1
,
x
2
,
…
,
x
n
and
y
1
,
y
2
,
…
,
y
n
y_1, y_2, …,y_n
y
1
,
y
2
,
…
,
y
n
be positive reals such that
x
1
+
x
2
+
.
.
+
x
n
≥
y
i
≥
x
i
2
x_1 + x_2 +.. + x_n \ge y_i \ge x^2_i
x
1
+
x
2
+
..
+
x
n
≥
y
i
≥
x
i
2
for all
i
=
1
,
2
,
…
,
n
i = 1, 2, …, n
i
=
1
,
2
,
…
,
n
. Prove that
x
1
x
1
y
1
+
x
2
+
+
x
2
x
2
y
2
+
x
3
+
.
.
.
+
x
n
x
n
y
n
+
x
1
>
1
2
n
.
\frac{x_1}{x_1y_1 + x_2}+ + \frac{x_2}{x_2y_2 + x_3} + ...+ \frac{x_n}{x_ny_n + x_1}> \frac{1}{2n}.
x
1
y
1
+
x
2
x
1
+
+
x
2
y
2
+
x
3
x
2
+
...
+
x
n
y
n
+
x
1
x
n
>
2
n
1
.
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