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Putnam
1947 Putnam
B1
Putnam 1947 B1
Putnam 1947 B1
Source: Putnam 1947
April 3, 2022
Putnam
function
trigonometry
Problem Statement
Let
f
(
x
)
f(x)
f
(
x
)
be a function such that
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
and for
x
≥
1
x \geq 1
x
≥
1
f
′
(
x
)
=
1
x
2
+
f
(
x
)
2
.
f'(x)= \frac{1}{x^2 +f(x)^{2}}.
f
′
(
x
)
=
x
2
+
f
(
x
)
2
1
.
Prove that
lim
x
→
∞
f
(
x
)
\lim_{x\to \infty} f(x)
x
→
∞
lim
f
(
x
)
exists and is less than
1
+
π
4
.
1+ \frac{\pi}{4}.
1
+
4
π
.
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