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International Contests
Pan African
2002 Pan African
6
Inequality with sum 1
Inequality with sum 1
Source: Pan African 2002
August 20, 2005
inequalities
induction
Problem Statement
If
a
1
≥
a
2
≥
⋯
≥
a
n
≥
0
a_1 \geq a_2 \geq \cdots \geq a_n \geq 0
a
1
≥
a
2
≥
⋯
≥
a
n
≥
0
and
a
1
+
a
2
+
⋯
+
a
n
=
1
a_1+a_2+\cdots+a_n=1
a
1
+
a
2
+
⋯
+
a
n
=
1
, then prove:
a
1
2
+
3
a
2
2
+
5
a
3
2
+
⋯
+
(
2
n
−
1
)
a
n
2
≤
1
a_1^2+3a_2^2+5a_3^2+ \cdots +(2n-1)a_n^2 \leq 1
a
1
2
+
3
a
2
2
+
5
a
3
2
+
⋯
+
(
2
n
−
1
)
a
n
2
≤
1
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