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Putnam
1987 Putnam
B6
B6
Part of
1987 Putnam
Problems
(1)
Putnam 1987 B6
Source:
8/5/2019
Let
F
F
F
be the field of
p
2
p^2
p
2
elements, where
p
p
p
is an odd prime. Suppose
S
S
S
is a set of
(
p
2
−
1
)
/
2
(p^2-1)/2
(
p
2
−
1
)
/2
distinct nonzero elements of
F
F
F
with the property that for each
a
≠
0
a\neq 0
a
=
0
in
F
F
F
, exactly one of
a
a
a
and
−
a
-a
−
a
is in
S
S
S
. Let
N
N
N
be the number of elements in the intersection
S
∩
{
2
a
:
a
∈
S
}
S \cap \{2a: a \in S\}
S
∩
{
2
a
:
a
∈
S
}
. Prove that
N
N
N
is even.
Putnam