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International Contests
IMO Longlists
1990 IMO Longlists
59
59
Part of
1990 IMO Longlists
Problems
(1)
Inequality on eight variables - ILL 1990 NET2
Source:
9/18/2010
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
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≤
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≤
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≤
a
7
≤
a
8
. Let
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=
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+
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+
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+
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8
x = \frac{ a_1 + a_2 + \cdots + a_7 + a_8}{8}
x
=
8
a
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+
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⋯
+
a
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+
a
8
,
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=
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y = \frac{ a_1^2 + a_2^2 + \cdots + a_7^2 + a_8^2}{8}
y
=
8
a
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+
a
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2
+
⋯
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a
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+
a
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2
. Prove that
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≤
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−
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≤
4
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−
x
2
.
2 \sqrt{y-x^2} \leq a_8 - a_1 \leq 4 \sqrt{y-x^2}.
2
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≤
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−
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≤
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.
inequalities
inequalities unsolved