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Canadian Open Math Challenge
2017 Canadian Open Math Challenge
C4
C4
Part of
2017 Canadian Open Math Challenge
Problems
(1)
2017 COMC C4
Source:
10/12/2018
Source: 2017 Canadian Open Math Challenge, Problem C4 —-- Let n be a positive integer and
S
n
=
{
1
,
2
,
.
.
.
,
2
n
−
1
,
2
n
}
S_n = \{1, 2, . . . , 2n - 1, 2n\}
S
n
=
{
1
,
2
,
...
,
2
n
−
1
,
2
n
}
. A perfect pairing of
S
n
S_n
S
n
is defined to be a partitioning of the
2
n
2n
2
n
numbers into
n
n
n
pairs, such that the sum of the two numbers in each pair is a perfect square. For example, if
n
=
4
n = 4
n
=
4
, then a perfect pairing of
S
4
S_4
S
4
is
(
1
,
8
)
,
(
2
,
7
)
,
(
3
,
6
)
,
(
4
,
5
)
(1, 8),(2, 7),(3, 6),(4, 5)
(
1
,
8
)
,
(
2
,
7
)
,
(
3
,
6
)
,
(
4
,
5
)
. It is not necessary for each pair to sum to the same perfect square. (a) Show that
S
8
S_8
S
8
has at least one perfect pairing. (b) Show that
S
5
S_5
S
5
does not have any perfect pairings. (c) Prove or disprove: there exists a positive integer
n
n
n
for which
S
n
S_n
S
n
has at least
2017
2017
2017
different perfect pairings. (Two pairings that are comprised of the same pairs written in a different order are considered the same pairing.)
Comc
2017 COMC