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India Contests
Regional Mathematical Olympiad
2000 India Regional Mathematical Olympiad
3
An inequality - good
An inequality - good
Source: Indian RMO 2000 Problem 3
October 26, 2005
inequalities
integration
calculus
induction
Problem Statement
Suppose
{
x
n
}
n
≥
1
\{ x_n \}_{n\geq 1}
{
x
n
}
n
≥
1
is a sequence of positive real numbers such that
x
1
≥
x
2
≥
x
3
…
≥
x
n
…
x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots
x
1
≥
x
2
≥
x
3
…
≥
x
n
…
, and for all
n
n
n
x
1
1
+
x
4
2
+
x
9
3
+
…
+
x
n
2
n
≤
1.
\frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 .
1
x
1
+
2
x
4
+
3
x
9
+
…
+
n
x
n
2
≤
1.
Show that for all
k
k
k
x
1
1
+
x
2
2
+
…
+
x
k
k
≤
3.
\frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3.
1
x
1
+
2
x
2
+
…
+
k
x
k
≤
3.
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