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Putnam
Putnam 1939
A7
A7
Part of
Putnam 1939
Problems
(1)
Putnam 1939 A7
Source:
8/20/2021
Do either
(
1
)
(1)
(
1
)
or
(
2
)
(2)
(
2
)
:
(
1
)
(1)
(
1
)
Let
C
a
C_a
C
a
be the curve
(
y
−
a
2
)
2
=
x
2
(
a
2
−
x
2
)
.
(y - a^2)^2 = x^2(a^2 - x^2).
(
y
−
a
2
)
2
=
x
2
(
a
2
−
x
2
)
.
Find the curve which touches all
C
a
C_a
C
a
for
a
>
0.
a > 0.
a
>
0.
Sketch the solution and at least two of the
C
a
.
C_a.
C
a
.
(
2
)
(2)
(
2
)
Given that
(
1
−
h
x
)
−
1
(
1
−
k
x
)
−
1
=
∑
i
≥
0
a
i
x
i
,
(1 - hx)^{-1}(1 - kx)^{-1} = \sum_{i\geq0}a_i x^i,
(
1
−
h
x
)
−
1
(
1
−
k
x
)
−
1
=
∑
i
≥
0
a
i
x
i
,
prove that
(
1
+
h
k
x
)
(
1
−
h
k
x
)
−
1
(
1
−
h
2
x
)
−
1
(
1
−
k
2
x
)
−
1
=
∑
i
≥
0
a
i
2
x
i
.
(1 + hkx)(1 - hkx)^{-1}(1 - h^2x)^{-1}(1 - k^2x)^{-1} = \sum_{i\geq0} a_i^2 x^i.
(
1
+
hk
x
)
(
1
−
hk
x
)
−
1
(
1
−
h
2
x
)
−
1
(
1
−
k
2
x
)
−
1
=
∑
i
≥
0
a
i
2
x
i
.
Putnam