MathDB
Problems
Contests
Undergraduate contests
Putnam
2012 Putnam
5
Putnam 2012 B5
Putnam 2012 B5
Source:
December 3, 2012
Putnam
function
logarithms
calculus
college contests
Problem Statement
Prove that, for any two bounded functions
g
1
,
g
2
:
R
→
[
1
,
∞
)
,
g_1,g_2 : \mathbb{R}\to[1,\infty),
g
1
,
g
2
:
R
→
[
1
,
∞
)
,
there exist functions
h
1
,
h
2
:
R
→
R
h_1,h_2 : \mathbb{R}\to\mathbb{R}
h
1
,
h
2
:
R
→
R
such that for every
x
∈
R
,
x\in\mathbb{R},
x
∈
R
,
sup
s
∈
R
(
g
1
(
s
)
x
g
2
(
s
)
)
=
max
t
∈
R
(
x
h
1
(
t
)
+
h
2
(
t
)
)
.
\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).
s
∈
R
sup
(
g
1
(
s
)
x
g
2
(
s
)
)
=
t
∈
R
max
(
x
h
1
(
t
)
+
h
2
(
t
)
)
.
Back to Problems
View on AoPS