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Ireland National Math Olympiad
2014 Irish Math Olympiad
5
Irish Mathematical Olympiad 2014 (2)
Irish Mathematical Olympiad 2014 (2)
Source: Paper 1 , Problem 5
September 6, 2014
inequalities proposed
inequalities
Problem Statement
Suppose
a
1
,
a
2
,
…
,
a
n
>
0
a_1,a_2,\ldots,a_n>0
a
1
,
a
2
,
…
,
a
n
>
0
, where
n
>
1
n>1
n
>
1
and
∑
i
=
1
n
a
i
=
1
\sum_{i=1}^{n}a_i=1
∑
i
=
1
n
a
i
=
1
. For each
i
=
1
,
2
,
…
,
n
i=1,2,\ldots,n
i
=
1
,
2
,
…
,
n
, let
b
i
=
a
i
2
∑
j
=
1
n
a
j
2
b_i=\frac{a^2_i}{\sum\limits_{j=1}^{n}a^2_j}
b
i
=
j
=
1
∑
n
a
j
2
a
i
2
. Prove that
∑
i
=
1
n
a
i
1
−
a
i
≤
∑
i
=
1
n
b
i
1
−
b
i
.
\sum_{i=1}^{n}\frac{a_i}{1-a_i}\le \sum_{i=1}^{n}\frac{b_i}{1-b_i} .
i
=
1
∑
n
1
−
a
i
a
i
≤
i
=
1
∑
n
1
−
b
i
b
i
.
When does equality occur ?
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