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Switzerland Team Selection Test
2019 Switzerland Team Selection Test
11
Inequality with All Means Except the QM
Inequality with All Means Except the QM
Source: Swiss TST 2019 P11
May 12, 2020
inequalities
Problem Statement
Let
n
n
n
be a positive integer. Determine whether there exists a positive real number
ϵ
>
0
\epsilon >0
ϵ
>
0
(depending on
n
n
n
) such that for all positive real numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots ,x_n
x
1
,
x
2
,
…
,
x
n
, the inequality
x
1
x
2
…
x
n
n
≤
(
1
−
ϵ
)
x
1
+
x
2
+
⋯
+
x
n
n
+
ϵ
n
1
x
1
+
1
x
2
+
⋯
+
1
x
n
,
\sqrt[n]{x_1x_2\dots x_n}\leq (1-\epsilon)\frac{x_1+x_2+\dots+x_n}{n}+\epsilon \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\dots +\frac{1}{x_n}},
n
x
1
x
2
…
x
n
≤
(
1
−
ϵ
)
n
x
1
+
x
2
+
⋯
+
x
n
+
ϵ
x
1
1
+
x
2
1
+
⋯
+
x
n
1
n
,
holds.
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