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Baltic Way
1997 Baltic Way
4
Arithmetic mean a of x_i
Arithmetic mean a of x_i
Source: Baltic Way 1997
January 28, 2011
inequalities
inequalities proposed
Problem Statement
Prove that the arithmetic mean
a
a
a
of
x
1
,
…
,
x
n
x_1,\ldots ,x_n
x
1
,
…
,
x
n
satisfies
(
x
1
−
a
)
2
+
…
+
(
x
n
−
a
)
2
≤
1
2
(
∣
x
1
−
a
∣
+
…
+
∣
x
n
−
a
∣
)
2
(x_1-a)^2+\ldots +(x_n-a)^2\le \frac{1}{2}(|x_1-a|+\ldots +|x_n-a|)^2
(
x
1
−
a
)
2
+
…
+
(
x
n
−
a
)
2
≤
2
1
(
∣
x
1
−
a
∣
+
…
+
∣
x
n
−
a
∣
)
2
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