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Putnam
2023 Putnam
A3
A3
Part of
2023 Putnam
Problems
(1)
2023 Putnam A3
Source:
12/3/2023
Determine the smallest positive real number
r
r
r
such that there exist differentiable functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
and
g
:
R
→
R
g: \mathbb{R} \rightarrow \mathbb{R}
g
:
R
→
R
satisfying (a)
f
(
0
)
>
0
f(0)>0
f
(
0
)
>
0
, (b)
g
(
0
)
=
0
g(0)=0
g
(
0
)
=
0
, (c)
∣
f
′
(
x
)
∣
≤
∣
g
(
x
)
∣
\left|f^{\prime}(x)\right| \leq|g(x)|
∣
f
′
(
x
)
∣
≤
∣
g
(
x
)
∣
for all
x
x
x
, (d)
∣
g
′
(
x
)
∣
≤
∣
f
(
x
)
∣
\left|g^{\prime}(x)\right| \leq|f(x)|
∣
g
′
(
x
)
∣
≤
∣
f
(
x
)
∣
for all
x
x
x
, and (e)
f
(
r
)
=
0
f(r)=0
f
(
r
)
=
0
.
Putnam
Putnam 2023