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Putnam
1992 Putnam
B3
B3
Part of
1992 Putnam
Problems
(1)
Putnam 1992 B3
Source: Putnam 1992
7/18/2022
For any pair
(
x
,
y
)
(x,y)
(
x
,
y
)
of real numbers, a sequence
(
a
n
(
x
,
y
)
)
(a_{n}(x,y))
(
a
n
(
x
,
y
))
is defined as follows:
a
0
(
x
,
y
)
=
x
,
a
n
+
1
(
x
,
y
)
=
a
n
(
x
,
y
)
2
+
y
2
2
for
n
≥
0
a_{0}(x,y)=x, \;\;\;\; a_{n+1}(x,y) =\frac{a_{n}(x,y)^{2} +y^2 }{2} \;\, \text{for}\, n\geq 0
a
0
(
x
,
y
)
=
x
,
a
n
+
1
(
x
,
y
)
=
2
a
n
(
x
,
y
)
2
+
y
2
for
n
≥
0
Find the area of the region
{
(
x
,
y
)
∈
R
2
∣
(
a
n
(
x
,
y
)
)
converges
}
\{(x,y)\in \mathbb{R}^{2} \, |\, (a_{n}(x,y)) \,\, \text{converges} \}
{(
x
,
y
)
∈
R
2
∣
(
a
n
(
x
,
y
))
converges
}
.
Putnam
geometry
Sequence
area