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Serbia Contests
Serbia Team Selection Test
1980 Yugoslav Team Selection Test
Problem 3
Problem 3
Part of
1980 Yugoslav Team Selection Test
Problems
(1)
fractional recurrence, output is integer
Source: Yugoslav TST 1980 P3
5/29/2021
A sequence
(
x
n
)
(x_n)
(
x
n
)
satisfies
x
n
+
1
=
x
n
2
+
a
x
n
−
1
x_{n+1}=\frac{x_n^2+a}{x_{n-1}}
x
n
+
1
=
x
n
−
1
x
n
2
+
a
for all
n
∈
N
n\in\mathbb N
n
∈
N
. Prove that if
x
0
,
x
1
x_0,x_1
x
0
,
x
1
, and
x
0
2
+
x
1
2
+
a
x
0
x
1
\frac{x_0^2+x_1^2+a}{x_0x_1}
x
0
x
1
x
0
2
+
x
1
2
+
a
are integers, then all the terms of sequence
(
x
n
)
(x_n)
(
x
n
)
are integers.
number theory
Sequences
recurrence relation