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Putnam
2021 Putnam
B3
B3
Part of
2021 Putnam
Problems
(1)
Putnam 2021 B3
Source:
12/5/2021
Let
h
(
x
,
y
)
h(x,y)
h
(
x
,
y
)
be a real-valued function that is twice continuously differentiable throughout
R
2
\mathbb{R}^2
R
2
, and define
ρ
(
x
,
y
)
=
y
h
x
−
x
h
y
.
\rho (x,y)=yh_x -xh_y .
ρ
(
x
,
y
)
=
y
h
x
−
x
h
y
.
Prove or disprove: For any positive constants
d
d
d
and
r
r
r
with
d
>
r
d>r
d
>
r
, there is a circle
S
S
S
of radius
r
r
r
whose center is a distance
d
d
d
away from the origin such that the integral of
ρ
\rho
ρ
over the interior of
S
S
S
is zero.
Putnam
Putnam 2021