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2016 Korea National Olympiad
6
Chebyshev with Max
Chebyshev with Max
Source: 2016 KMO Senior #6
November 12, 2016
inequalities
Problem Statement
For a positive integer
n
n
n
, there are
n
n
n
positive reals
a
1
≥
a
2
≥
a
3
⋯
≥
a
n
a_1 \ge a_2 \ge a_3 \cdots \ge a_n
a
1
≥
a
2
≥
a
3
⋯
≥
a
n
. For all positive reals
b
1
,
b
2
,
⋯
b
n
b_1, b_2, \cdots b_n
b
1
,
b
2
,
⋯
b
n
, prove the following inequality.
a
1
b
1
+
a
2
b
2
+
⋯
+
a
n
b
n
a
1
+
a
2
+
⋯
a
n
≤
max
{
b
1
1
,
b
1
+
b
2
2
,
⋯
,
b
1
+
b
2
+
⋯
+
b
n
n
}
\frac{a_1b_1+a_2b_2 + \cdots +a_nb_n}{a_1+a_2+ \cdots a_n} \le \text{max}\{ \frac{b_1}{1}, \frac{b_1+b_2}{2}, \cdots, \frac{b_1+b_2+ \cdots +b_n}{n} \}
a
1
+
a
2
+
⋯
a
n
a
1
b
1
+
a
2
b
2
+
⋯
+
a
n
b
n
≤
max
{
1
b
1
,
2
b
1
+
b
2
,
⋯
,
n
b
1
+
b
2
+
⋯
+
b
n
}
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