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Putnam
1965 Putnam
A6
A6
Part of
1965 Putnam
Problems
(1)
Putnam 1965 A6
Source:
9/27/2020
In the plane with orthogonal Cartesian coordinates
x
x
x
and
y
y
y
, prove that the line whose equation is
u
x
+
v
y
=
1
ux+vy = 1
ux
+
v
y
=
1
will be tangent to the cirve
x
m
+
y
m
=
1
x^m+y^m=1
x
m
+
y
m
=
1
(where
m
>
1
m>1
m
>
1
) if and only if
u
n
+
v
n
=
1
u^n + v^n = 1
u
n
+
v
n
=
1
and
m
−
1
+
n
−
1
=
1
m^{-1} + n^{-1} = 1
m
−
1
+
n
−
1
=
1
.
Putnam