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National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2018 JBMO TST-Turkey
5
a+2018 is a perfect square
a+2018 is a perfect square
Source: 2018 JBMO TST - Turkey, P5
March 27, 2020
NumberTheory
Problem Statement
Let
a
1
,
a
2
,
.
.
.
,
a
1000
a_1, a_2, ... , a_{1000}
a
1
,
a
2
,
...
,
a
1000
be a sequence of integers such that
a
1
=
3
,
a
2
=
7
a_1=3, a_2=7
a
1
=
3
,
a
2
=
7
and for all
n
=
2
,
3
,
.
.
.
,
999
n=2, 3, ... , 999
n
=
2
,
3
,
...
,
999
a
n
+
1
−
a
n
=
4
(
a
1
+
a
2
)
(
a
2
+
a
3
)
.
.
.
(
a
n
−
1
+
a
n
)
a_{n+1}-a_n=4(a_1+a_2)(a_2+a_3) ... (a_{n-1}+a_n)
a
n
+
1
−
a
n
=
4
(
a
1
+
a
2
)
(
a
2
+
a
3
)
...
(
a
n
−
1
+
a
n
)
. Find the number of indices
1
≤
n
≤
1000
1\leq n\leq 1000
1
≤
n
≤
1000
for which
a
n
+
2018
a_n+2018
a
n
+
2018
is a perfect square.
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