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Putnam
2000 Putnam
4
4
Part of
2000 Putnam
Problems
(2)
Putnam 2000 A4
Source:
9/6/2011
Show that the improper integral
lim
B
→
∞
∫
0
B
sin
(
x
)
sin
(
x
2
)
d
x
\lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx
B
→
∞
lim
∫
0
B
sin
(
x
)
sin
(
x
2
)
d
x
converges.
Putnam
calculus
integration
limit
trigonometry
function
college contests
Putnam 2000 B4
Source:
9/6/2011
Let
f
(
x
)
f(x)
f
(
x
)
be a continuous function such that
f
(
2
x
2
−
1
)
=
2
x
f
(
x
)
f(2x^2-1)=2xf(x)
f
(
2
x
2
−
1
)
=
2
x
f
(
x
)
for all
x
x
x
. Show that
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
for
−
1
≤
x
≤
1
-1\le x \le 1
−
1
≤
x
≤
1
.
Putnam
function
trigonometry
induction
college contests