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Putnam
2020 Putnam
B1
B1
Part of
2020 Putnam
Problems
(1)
Putnam 2020 B1
Source: 81st William Lowell Putnam Competition
2/22/2021
For a positive integer
n
n
n
, define
d
(
n
)
d(n)
d
(
n
)
to be the sum of the digits of
n
n
n
when written in binary (for example,
d
(
13
)
=
1
+
1
+
0
+
1
=
3
d(13)=1+1+0+1=3
d
(
13
)
=
1
+
1
+
0
+
1
=
3
). Let
S
=
∑
k
=
1
2020
(
−
1
)
d
(
k
)
k
3
.
S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3.
S
=
k
=
1
∑
2020
(
−
1
)
d
(
k
)
k
3
.
Determine
S
S
S
modulo
2020
2020
2020
.
Putnam
abstract algebra
Putnam 2020