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Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2022 Bangladesh Mathematical Olympiad
6
6
Part of
2022 Bangladesh Mathematical Olympiad
Problems
(1)
Sum of remainder pairs greater than 2017=N
Source: BdMO 2022 Secondary P6
4/12/2022
About
5
5
5
years ago, Joydip was researching on the number
2017
2017
2017
. He understood that
2017
2017
2017
is a prime number. Then he took two integers
a
,
b
a,b
a
,
b
such that
0
<
a
,
b
<
2017
0<a,b <2017
0
<
a
,
b
<
2017
and
a
+
b
≠
2017.
a+b\neq 2017.
a
+
b
=
2017.
He created two sequences
A
1
,
A
2
,
…
,
A
2016
A_1,A_2,\dots ,A_{2016}
A
1
,
A
2
,
…
,
A
2016
and
B
1
,
B
2
,
…
,
B
2016
B_1,B_2,\dots, B_{2016}
B
1
,
B
2
,
…
,
B
2016
where
A
k
A_k
A
k
is the remainder upon dividing
a
k
ak
ak
by
2017
2017
2017
, and
B
k
B_k
B
k
is the remainder upon dividing
b
k
bk
bk
by
2017.
2017.
2017.
Among the numbers
A
1
+
B
1
,
A
2
+
B
2
,
…
A
2016
+
B
2016
A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016}
A
1
+
B
1
,
A
2
+
B
2
,
…
A
2016
+
B
2016
count of those that are greater than
2017
2017
2017
is
N
N
N
. Prove that
N
=
1008.
N=1008.
N
=
1008.
number theory