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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2019 China Second Round Olympiad
1
Inequality with 100 variables.
Inequality with 100 variables.
Source: 2019 China Second Round P1 (Paper B)
September 8, 2019
inequalities
Problem Statement
Suppose that
a
1
,
a
2
,
⋯
,
a
100
∈
R
+
a_1,a_2,\cdots,a_{100}\in\mathbb{R}^+
a
1
,
a
2
,
⋯
,
a
100
∈
R
+
such that
a
i
≥
a
101
−
i
(
i
=
1
,
2
,
⋯
,
50
)
.
a_i\geq a_{101-i}\,(i=1,2,\cdots,50).
a
i
≥
a
101
−
i
(
i
=
1
,
2
,
⋯
,
50
)
.
Let
x
k
=
k
a
k
+
1
a
1
+
a
2
+
⋯
+
a
k
(
k
=
1
,
2
,
⋯
,
99
)
.
x_k=\frac{ka_{k+1}}{a_1+a_2+\cdots+a_k}\,(k=1,2,\cdots,99).
x
k
=
a
1
+
a
2
+
⋯
+
a
k
k
a
k
+
1
(
k
=
1
,
2
,
⋯
,
99
)
.
Prove that
x
1
x
2
2
⋯
x
99
99
≤
1.
x_1x_2^2\cdots x_{99}^{99}\leq 1.
x
1
x
2
2
⋯
x
99
99
≤
1.
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