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National and Regional Contests
Korea Contests
Korea National Olympiad
2005 Korea National Olympiad
6
19th kmo #6
19th kmo #6
Source: KMO round 2, problem 6
February 3, 2006
inequalities unsolved
inequalities
Problem Statement
Real numbers
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,
x
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x
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⋯
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x
n
x_1, x_2, x_3, \cdots , x_n
x
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⋯
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x
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satisfy
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x
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2
=
1
x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2 = 1
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=
1
. Show that
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x
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1
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x
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⋯
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1
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x
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<
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2
.
\frac{x_1}{1+x_1^2}+\frac{x_2}{1+x_1^2+x_2^2}+\cdots+\frac{x_n}{1+ x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2} < \sqrt{\frac n2} .
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x
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2
x
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<
2
n
.
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