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Putnam
1949 Putnam
B6
B6
Part of
1949 Putnam
Problems
(1)
Putnam 1949 B6
Source: Putnam 1949
3/20/2022
Let
C
C
C
be a closed convex curve with a continuously turning tangent and let
O
O
O
be a point inside
C
.
C.
C
.
For each point
P
P
P
on
C
C
C
we define
T
(
P
)
T(P)
T
(
P
)
as follows: Draw the tangent to
C
C
C
at
P
P
P
and from
O
O
O
drop the perpendicular to that tangent. Then
T
(
P
)
T(P)
T
(
P
)
is the point at which
C
C
C
intersects this perpendicular. Starting now with a point
P
0
P_{0}
P
0
on
C
C
C
, define points
P
n
P_n
P
n
by
P
n
=
T
(
P
n
−
1
)
.
P_n =T(P_{n-1}).
P
n
=
T
(
P
n
−
1
)
.
Prove that the points
P
n
P_{n}
P
n
approach a limit and characterize all possible limit points. (You may assume that
T
T
T
is continuous.)
Putnam
geometry
tangent