MathDB
Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2017 China Western Mathematical Olympiad
8
2017 China Western Mathematical Olympiad Q8
2017 China Western Mathematical Olympiad Q8
Source: Sichuan Nanchong
August 14, 2017
inequalities
maximum and minimum
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
n
>
0
a_1,a_2,\cdots,a_n>0
a
1
,
a
2
,
⋯
,
a
n
>
0
(
n
≥
2
)
(n\geq 2)
(
n
≥
2
)
. Prove that
∑
i
=
1
n
m
a
x
{
a
1
,
a
2
,
⋯
,
a
i
}
⋅
m
i
n
{
a
i
,
a
i
+
1
,
⋯
,
a
n
}
≤
n
2
n
−
1
∑
i
=
1
n
a
i
2
\sum_{i=1}^n max\{a_1,a_2,\cdots,a_i \} \cdot min \{a_i,a_{i+1},\cdots,a_n\}\leq \frac{n}{2\sqrt{n-1}}\sum_{i=1}^n a^2_i
i
=
1
∑
n
ma
x
{
a
1
,
a
2
,
⋯
,
a
i
}
⋅
min
{
a
i
,
a
i
+
1
,
⋯
,
a
n
}
≤
2
n
−
1
n
i
=
1
∑
n
a
i
2
Back to Problems
View on AoPS