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Problem 4
Inequality for all real numbers
Inequality for all real numbers
Source: Swiss IMO TST 2016. Problem 4
July 27, 2017
inequalities
Problem Statement
Find all integers
n
≥
1
n \geq 1
n
≥
1
such that for all
x
1
,
.
.
.
,
x
n
∈
R
x_1,...,x_n \in \mathbb{R}
x
1
,
...
,
x
n
∈
R
the following inequality is satisfied
(
x
1
n
+
.
.
.
+
x
n
n
n
−
x
1
.
.
.
.
x
n
)
(
x
1
+
.
.
.
+
x
n
)
≥
0
\left(\frac{x_1^n+...+x_n^n}{n}-x_1....x_n\right)\left(x_1+...+x_n\right) \geq 0
(
n
x
1
n
+
...
+
x
n
n
−
x
1
....
x
n
)
(
x
1
+
...
+
x
n
)
≥
0
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