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Putnam
1972 Putnam
A6
A6
Part of
1972 Putnam
Problems
(1)
Putnam 1972 A6
Source: Putnam 1972
2/17/2022
Let
f
f
f
be an integrable real-valued function on the closed interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
such that
∫
0
1
x
m
f
(
x
)
d
x
=
{
0
for
m
=
0
,
1
,
…
,
n
−
1
;
1
for
m
=
n
.
\int_{0}^{1} x^{m}f(x) dx=\begin{cases} 0 \;\; \text{for}\; m=0,1,\ldots,n-1;\\ 1\;\; \text{for}\; m=n. \end{cases}
∫
0
1
x
m
f
(
x
)
d
x
=
{
0
for
m
=
0
,
1
,
…
,
n
−
1
;
1
for
m
=
n
.
Show that
∣
f
(
x
)
∣
≥
2
n
(
n
+
1
)
|f(x)|\geq2^{n}(n+1)
∣
f
(
x
)
∣
≥
2
n
(
n
+
1
)
on a set of positive measure.
Putnam