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Problems
Contests
International Contests
APMO
1990 APMO
2
2
Part of
1990 APMO
Problems
(1)
Sum of product
Source: APMO 1990
3/11/2006
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
⋯
\cdots
⋯
,
a
n
a_n
a
n
be positive real numbers, and let
S
k
S_k
S
k
be the sum of the products of
a
1
a_1
a
1
,
a
2
a_2
a
2
,
⋯
\cdots
⋯
,
a
n
a_n
a
n
taken
k
k
k
at a time. Show that
S
k
S
n
−
k
≥
(
n
k
)
2
a
1
a
2
⋯
a
n
S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n
S
k
S
n
−
k
≥
(
k
n
)
2
a
1
a
2
⋯
a
n
for
k
=
1
k = 1
k
=
1
,
2
2
2
,
⋯
\cdots
⋯
,
n
−
1
n - 1
n
−
1
.
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algebra
polynomial
inequalities unsolved
inequalities