MathDB
Problems
Contests
National and Regional Contests
Colombia Contests
Colombia Team Selection Test
2005 Colombia Team Selection Test
4
Nice inequality with means
Nice inequality with means
Source: Colombia TSt
June 7, 2005
inequalities
induction
inequalities proposed
Problem Statement
1. Prove the following inequality for positive reals
a
1
,
a
2
.
.
.
,
a
n
a_1,a_2...,a_n
a
1
,
a
2
...
,
a
n
and
b
1
,
b
2
.
.
.
,
b
n
b_1,b_2...,b_n
b
1
,
b
2
...
,
b
n
:
(
∑
a
i
)
(
∑
b
i
)
≥
(
∑
a
i
+
b
i
)
(
∑
a
i
b
i
a
i
+
b
i
)
(\sum a_i)(\sum b_i)\geq (\sum a_i+b_i)(\sum\frac{a_ib_i}{a_i+b_i})
(
∑
a
i
)
(
∑
b
i
)
≥
(
∑
a
i
+
b
i
)
(
∑
a
i
+
b
i
a
i
b
i
)
Back to Problems
View on AoPS