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South Africa National Olympiad
2010 South africa National Olympiad
4
Sum of (x_k)/(sqrt k)
Sum of (x_k)/(sqrt k)
Source: South African MO 2010 Q4
May 26, 2012
inequalities unsolved
inequalities
Problem Statement
Given
n
n
n
positive real numbers satisfying
x
1
≥
x
2
≥
⋯
≥
x
n
≥
0
x_1 \ge x_2 \ge \cdots \ge x_n \ge 0
x
1
≥
x
2
≥
⋯
≥
x
n
≥
0
and
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
1
x_1^2+x_2^2+\cdots+x_n^2=1
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
1
, prove that
x
1
1
+
x
2
2
+
⋯
+
x
n
n
≥
1.
\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.
1
x
1
+
2
x
2
+
⋯
+
n
x
n
≥
1.
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