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Putnam
1987 Putnam
A2
A2
Part of
1987 Putnam
Problems
(1)
Putnam 1987 A2
Source:
8/5/2019
The sequence of digits
123456789101112131415161718192021
…
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \dots
123456789101112131415161718192021
…
is obtained by writing the positive integers in order. If the
1
0
n
10^n
1
0
n
-th digit in this sequence occurs in the part of the sequence in which the
m
m
m
-digit numbers are placed, define
f
(
n
)
f(n)
f
(
n
)
to be
m
m
m
. For example,
f
(
2
)
=
2
f(2)=2
f
(
2
)
=
2
because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof,
f
(
1987
)
f(1987)
f
(
1987
)
.
Putnam