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National and Regional Contests
Korea Contests
Korea Winter Program Practice Test
2016 Korea Winter Program Practice Test
2
Easy inequality
Easy inequality
Source: 2016 Korea Winter Camp 1st Test #6
January 25, 2016
inequalities
Problem Statement
Let
a
i
,
b
i
a_i, b_i
a
i
,
b
i
(
1
≤
i
≤
n
1 \le i \le n
1
≤
i
≤
n
,
n
≥
2
n \ge 2
n
≥
2
) be positive real numbers such that
∑
i
=
1
n
a
i
=
∑
i
=
1
n
b
i
\sum_{i=1}^n a_i = \sum_{i=1}^n b_i
∑
i
=
1
n
a
i
=
∑
i
=
1
n
b
i
.Prove that
∑
i
=
1
n
(
a
i
+
1
+
b
i
+
1
)
2
n
(
a
i
−
b
i
)
2
+
4
(
n
−
1
)
∑
j
=
1
n
a
j
b
j
≥
1
n
−
1
\sum_{i=1}^n \frac{(a_{i+1}+b_{i+1})^2}{n(a_i-b_i)^2+4(n-1)\sum_{j=1}^n a_jb_j} \ge \frac{1}{n-1}
∑
i
=
1
n
n
(
a
i
−
b
i
)
2
+
4
(
n
−
1
)
∑
j
=
1
n
a
j
b
j
(
a
i
+
1
+
b
i
+
1
)
2
≥
n
−
1
1
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