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China Girls Math Olympiad
2019 China Girls Math Olympiad
6
6
Part of
2019 China Girls Math Olympiad
Problems
(1)
Chinese Girls Mathematical Olympiad 2019, Problem 6
Source: China Girls Math Olympiad 2019 Day 2 P2
8/13/2019
Let
0
≤
x
1
≤
x
2
≤
⋯
≤
x
n
≤
1
0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1
0
≤
x
1
≤
x
2
≤
⋯
≤
x
n
≤
1
(
n
≥
2
)
.
(n\geq 2).
(
n
≥
2
)
.
Prove that
x
1
x
2
⋯
x
n
n
+
(
1
−
x
1
)
(
1
−
x
2
)
⋯
(
1
−
x
n
)
n
≤
1
−
(
x
1
−
x
n
)
2
n
.
\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.
n
x
1
x
2
⋯
x
n
+
n
(
1
−
x
1
)
(
1
−
x
2
)
⋯
(
1
−
x
n
)
≤
n
1
−
(
x
1
−
x
n
)
2
.
inequalities
algebra
China
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