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1.8
2022 Alg/NT Div 1 P8
2022 Alg/NT Div 1 P8
Source:
February 28, 2022
algebra
number theory
Problem Statement
Find the largest
c
>
0
c > 0
c
>
0
such that for all
n
≥
1
n \ge 1
n
≥
1
and
a
1
,
…
,
a
n
,
b
1
,
…
,
b
n
>
0
a_1,\dots,a_n, b_1,\dots, b_n > 0
a
1
,
…
,
a
n
,
b
1
,
…
,
b
n
>
0
we have
∑
j
=
1
n
a
j
4
≥
c
∑
k
=
1
n
(
∑
j
=
1
k
a
j
b
k
+
1
−
j
)
4
(
∑
j
=
1
k
b
j
2
j
!
)
2
\sum_{j=1}^n a_j^4 \ge c\sum_{k = 1}^n \frac{\left(\sum_{j=1}^k a_jb_{k+1-j}\right)^4}{\left(\sum_{j=1}^k b_j^2j!\right)^2}
j
=
1
∑
n
a
j
4
≥
c
k
=
1
∑
n
(
∑
j
=
1
k
b
j
2
j
!
)
2
(
∑
j
=
1
k
a
j
b
k
+
1
−
j
)
4
Proposed by Grant Yu
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