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Putnam
1978 Putnam
B6
B6
Part of
1978 Putnam
Problems
(1)
Putnam 1978 B6
Source: Putnam 1978
5/2/2022
Let
p
p
p
and
n
n
n
be positive integers. Suppose that the numbers
c
h
k
c_{hk}
c
hk
(
h
=
1
,
2
,
…
,
n
h=1,2,\ldots,n
h
=
1
,
2
,
…
,
n
;
k
=
1
,
2
,
…
,
p
h
k=1,2,\ldots,ph
k
=
1
,
2
,
…
,
p
h
) satisfy
0
≤
c
h
k
≤
1.
0 \leq c_{hk} \leq 1.
0
≤
c
hk
≤
1.
Prove that
(
∑
c
h
k
h
)
2
≤
2
p
∑
c
h
k
,
\left( \sum \frac{ c_{hk} }{h} \right)^2 \leq 2p \sum c_{hk} ,
(
∑
h
c
hk
)
2
≤
2
p
∑
c
hk
,
where each summation is over all admissible ordered pairs
(
h
,
k
)
.
(h,k).
(
h
,
k
)
.
Putnam
Summation
inequalities