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1990 IMO Longlists
59
Inequality on eight variables - ILL 1990 NET2
Inequality on eight variables - ILL 1990 NET2
Source:
September 18, 2010
inequalities
inequalities unsolved
Problem Statement
Given eight real numbers
a
1
≤
a
2
≤
⋯
≤
a
7
≤
a
8
a_1 \leq a_2 \leq \cdots \leq a_7 \leq a_8
a
1
≤
a
2
≤
⋯
≤
a
7
≤
a
8
. Let
x
=
a
1
+
a
2
+
⋯
+
a
7
+
a
8
8
x = \frac{ a_1 + a_2 + \cdots + a_7 + a_8}{8}
x
=
8
a
1
+
a
2
+
⋯
+
a
7
+
a
8
,
y
=
a
1
2
+
a
2
2
+
⋯
+
a
7
2
+
a
8
2
8
y = \frac{ a_1^2 + a_2^2 + \cdots + a_7^2 + a_8^2}{8}
y
=
8
a
1
2
+
a
2
2
+
⋯
+
a
7
2
+
a
8
2
. Prove that
2
y
−
x
2
≤
a
8
−
a
1
≤
4
y
−
x
2
.
2 \sqrt{y-x^2} \leq a_8 - a_1 \leq 4 \sqrt{y-x^2}.
2
y
−
x
2
≤
a
8
−
a
1
≤
4
y
−
x
2
.
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