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Putnam
2011 Putnam
B5
Putnam 2011 B5
Putnam 2011 B5
Source:
December 5, 2011
Putnam
integration
function
inequalities
calculus
Support
vector
Problem Statement
Let
a
1
,
a
2
,
…
a_1,a_2,\dots
a
1
,
a
2
,
…
be real numbers. Suppose there is a constant
A
A
A
such that for all
n
,
n,
n
,
∫
−
∞
∞
(
∑
i
=
1
n
1
1
+
(
x
−
a
i
)
2
)
2
d
x
≤
A
n
.
\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.
∫
−
∞
∞
(
i
=
1
∑
n
1
+
(
x
−
a
i
)
2
1
)
2
d
x
≤
A
n
.
Prove there is a constant
B
>
0
B>0
B
>
0
such that for all
n
,
n,
n
,
∑
i
,
j
=
1
n
(
1
+
(
a
i
−
a
j
)
2
)
≥
B
n
3
.
\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.
i
,
j
=
1
∑
n
(
1
+
(
a
i
−
a
j
)
2
)
≥
B
n
3
.
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