MathDB
Problems
Contests
Undergraduate contests
Putnam
2004 Putnam
A6
Putnam 2004 A6
Putnam 2004 A6
Source:
December 11, 2004
Putnam
function
integration
calculus
college contests
Problem Statement
Suppose that
f
(
x
,
y
)
f(x,y)
f
(
x
,
y
)
is a continuous real-valued function on the unit square
0
≤
x
≤
1
,
0
≤
y
≤
1.
0\le x\le1,0\le y\le1.
0
≤
x
≤
1
,
0
≤
y
≤
1.
Show that
∫
0
1
(
∫
0
1
f
(
x
,
y
)
d
x
)
2
d
y
+
∫
0
1
(
∫
0
1
f
(
x
,
y
)
d
y
)
2
d
x
\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx
∫
0
1
(
∫
0
1
f
(
x
,
y
)
d
x
)
2
d
y
+
∫
0
1
(
∫
0
1
f
(
x
,
y
)
d
y
)
2
d
x
≤
(
∫
0
1
∫
0
1
f
(
x
,
y
)
d
x
d
y
)
2
+
∫
0
1
∫
0
1
[
f
(
x
,
y
)
]
2
d
x
d
y
.
\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.
≤
(
∫
0
1
∫
0
1
f
(
x
,
y
)
d
x
d
y
)
2
+
∫
0
1
∫
0
1
[
f
(
x
,
y
)
]
2
d
x
d
y
.
Back to Problems
View on AoPS