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Putnam
1999 Putnam
3
Putnam 1999 B3
Putnam 1999 B3
Source:
October 28, 2012
Putnam
limit
algebra
polynomial
modular arithmetic
college contests
Putnam calculus
Problem Statement
Let
A
=
{
(
x
,
y
)
:
0
≤
x
,
y
<
1
}
.
A=\{(x,y): 0\le x,y < 1\}.
A
=
{(
x
,
y
)
:
0
≤
x
,
y
<
1
}
.
For
(
x
,
y
)
∈
A
,
(x,y)\in A,
(
x
,
y
)
∈
A
,
let
S
(
x
,
y
)
=
∑
1
2
≤
m
n
≤
2
x
m
y
n
,
S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,
S
(
x
,
y
)
=
2
1
≤
n
m
≤
2
∑
x
m
y
n
,
where the sum ranges over all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of positive integers satisfying the indicated inequalities. Evaluate
lim
(
x
,
y
)
→
(
1
,
1
)
,
(
x
,
y
)
∈
A
(
1
−
x
y
2
)
(
1
−
x
2
y
)
S
(
x
,
y
)
.
\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).
(
x
,
y
)
→
(
1
,
1
)
,
(
x
,
y
)
∈
A
lim
(
1
−
x
y
2
)
(
1
−
x
2
y
)
S
(
x
,
y
)
.
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