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KoMaL A Problems 2017/2018
A. 721
Inequality with positive reals summing to 1
Inequality with positive reals summing to 1
Source: KöMaL A. 721
April 13, 2018
inequalities
algebra
Problem Statement
Let
n
≥
2
n\ge 2
n
≥
2
be a positive integer, and suppose
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots ,a_n
a
1
,
a
2
,
⋯
,
a
n
are positive real numbers whose sum is
1
1
1
and whose squares add up to
S
S
S
. Prove that if
b
i
=
a
i
2
S
(
i
=
1
,
⋯
,
n
)
b_i=\tfrac{a^2_i}{S} \;(i=1,\cdots ,n)
b
i
=
S
a
i
2
(
i
=
1
,
⋯
,
n
)
, then for every
r
>
0
r>0
r
>
0
, we have
∑
i
=
1
n
a
i
(
1
−
a
i
)
r
≤
∑
i
=
1
n
b
i
(
1
−
b
i
)
r
.
\sum_{i=1}^n \frac{a_i}{{(1-a_i)}^r}\le \sum_{i=1}^n \frac{b_i}{{(1-b_i)}^r}.
i
=
1
∑
n
(
1
−
a
i
)
r
a
i
≤
i
=
1
∑
n
(
1
−
b
i
)
r
b
i
.
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