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Poland - Second Round
1975 Poland - Second Round
4
( sum a_i x_i^2 )^2 <= sum a_i x_i^4
( sum a_i x_i^2 )^2 <= sum a_i x_i^4
Source: Polish MO Second Round 1975 p4
September 8, 2024
algebra
inequalities
Problem Statement
Prove that the non-negative numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
(
n
=
1
,
2
,
…
n = 1, 2, \ldots
n
=
1
,
2
,
…
) satisfy the inequality
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
for any real numbers
(
∑
i
=
1
n
a
i
x
i
2
)
2
≤
∑
i
=
1
n
a
i
x
i
4
.
\left( \sum_{i=1}^n a_i x_i^2 \right)^2 \leq \sum_{i=1}^n a_i x_i^4.
(
i
=
1
∑
n
a
i
x
i
2
)
2
≤
i
=
1
∑
n
a
i
x
i
4
.
it is necessary and sufficient that
∑
i
=
1
n
a
i
≤
1
\sum_{i=1}^n a_i \leq 1
∑
i
=
1
n
a
i
≤
1
.
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