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MathLinks Contest 4th
6.3
0463 tetahedron 4th edition Round 6 p3
0463 tetahedron 4th edition Round 6 p3
Source:
May 7, 2021
inequalities
4th edition
algebra
Problem Statement
If
n
>
2
n>2
n
>
2
is an integer and
x
1
,
…
,
x
n
x_1, \ldots ,x_n
x
1
,
…
,
x
n
are positive reals such that
1
x
1
+
1
x
2
+
⋯
+
1
x
n
=
n
\frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n} = n
x
1
1
+
x
2
1
+
⋯
+
x
n
1
=
n
then the following inequality takes place
x
2
2
+
⋯
+
x
n
2
n
−
1
⋅
x
1
2
+
x
3
2
+
⋯
+
x
n
2
n
−
1
⋯
x
1
2
+
⋯
+
x
n
−
1
2
n
−
1
≥
(
x
1
2
+
.
.
.
+
x
n
2
n
)
n
−
1
.
\frac{x_2^2+\cdots+x_n^2}{n-1}\cdot \frac {x_1^2+x_3^2+\cdots +x_n^2} {n-1} \cdots \frac{x_1^2+\cdots+x_{n-1}^2}{n-1}\geq \left(\frac{x_1^2+...+x_n^2}{n}\right)^{n-1}.
n
−
1
x
2
2
+
⋯
+
x
n
2
⋅
n
−
1
x
1
2
+
x
3
2
+
⋯
+
x
n
2
⋯
n
−
1
x
1
2
+
⋯
+
x
n
−
1
2
≥
(
n
x
1
2
+
...
+
x
n
2
)
n
−
1
.
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