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Turkey Team Selection Test
2014 Turkey Team Selection Test
2
Natural Number Sequence
Natural Number Sequence
Source: Turkey TST 2014 Day 3 Problem 8
March 12, 2014
number theory proposed
number theory
Problem Statement
a
1
=
−
5
a_1=-5
a
1
=
−
5
,
a
2
=
−
6
a_2=-6
a
2
=
−
6
and for all
n
≥
2
n \geq 2
n
≥
2
the
(
a
n
)
∞
n
=
1
{(a_n)^\infty}_{n=1}
(
a
n
)
∞
n
=
1
sequence defined as,
a
n
+
1
=
a
n
+
(
a
1
+
1
)
(
2
a
2
+
1
)
(
3
a
3
+
1
)
⋯
(
(
n
−
1
)
a
n
−
1
+
1
)
(
(
n
2
+
n
)
a
n
+
2
n
+
1
)
)
.
a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)).
a
n
+
1
=
a
n
+
(
a
1
+
1
)
(
2
a
2
+
1
)
(
3
a
3
+
1
)
⋯
((
n
−
1
)
a
n
−
1
+
1
)
((
n
2
+
n
)
a
n
+
2
n
+
1
))
.
If a prime
p
p
p
divides
n
a
n
+
1
na_n+1
n
a
n
+
1
for a natural number n, prove that there is a integer
m
m
m
such that
m
2
≡
5
(
m
o
d
p
)
m^2\equiv5(modp)
m
2
≡
5
(
m
o
d
p
)
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