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Romanian Masters of Mathematics Collection
2019 Romanian Master of Mathematics Shortlist
N1
N1
Part of
2019 Romanian Master of Mathematics Shortlist
Problems
(1)
set A contains at least (p - 1)(q + 1)/8 pairs whose entries are both even
Source: 2019 RMM Shortlist N1
6/19/2020
Let
p
p
p
and
q
q
q
be relatively prime positive odd integers such that
1
<
p
<
q
1 < p < q
1
<
p
<
q
. Let
A
A
A
be a set of pairs of integers
(
a
,
b
)
(a, b)
(
a
,
b
)
, where
0
≤
a
≤
p
−
1
,
0
≤
b
≤
q
−
1
0 \le a \le p - 1, 0 \le b \le q - 1
0
≤
a
≤
p
−
1
,
0
≤
b
≤
q
−
1
, containing exactly one pair from each of the sets
{
(
a
,
b
)
,
(
a
+
1
,
b
+
1
)
}
,
{
(
a
,
q
−
1
)
,
(
a
+
1
,
0
)
}
,
{
(
p
−
1
,
b
)
,
(
0
,
b
+
1
)
}
\{(a, b),(a + 1, b + 1)\}, \{(a, q - 1), (a + 1, 0)\}, \{(p - 1,b),(0, b + 1)\}
{(
a
,
b
)
,
(
a
+
1
,
b
+
1
)}
,
{(
a
,
q
−
1
)
,
(
a
+
1
,
0
)}
,
{(
p
−
1
,
b
)
,
(
0
,
b
+
1
)}
whenever
0
≤
a
≤
p
−
2
0 \le a \le p - 2
0
≤
a
≤
p
−
2
and
0
≤
b
≤
q
−
2
0 \le b \le q - 2
0
≤
b
≤
q
−
2
. Show that
A
A
A
contains at least
(
p
−
1
)
(
q
+
1
)
/
8
(p - 1)(q + 1)/8
(
p
−
1
)
(
q
+
1
)
/8
pairs whose entries are both even.Agnijo Banerjee and Joe Benton, United Kingdom
number theory
Even