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Vietnam Contests
Vietnam Team Selection Test
2023 Vietnam Team Selection Test
4
4
Part of
2023 Vietnam Team Selection Test
Problems
(1)
Sequence NT
Source: Vietnam TST 2023 P4
4/14/2023
Given are two coprime positive integers
a
,
b
a, b
a
,
b
with
b
b
b
odd and
a
>
2
a>2
a
>
2
. The sequence
(
x
n
)
(x_n)
(
x
n
)
is defined by
x
0
=
2
,
x
1
=
a
x_0=2, x_1=a
x
0
=
2
,
x
1
=
a
and
x
n
+
2
=
a
x
n
+
1
+
b
x
n
x_{n+2}=ax_{n+1}+bx_n
x
n
+
2
=
a
x
n
+
1
+
b
x
n
for
n
≥
1
n \geq 1
n
≥
1
. Prove that:
a
)
a)
a
)
If
a
a
a
is even then there do not exist positive integers
m
,
n
,
p
m, n, p
m
,
n
,
p
such that
x
m
x
n
x
p
\frac{x_m} {x_nx_p}
x
n
x
p
x
m
is a positive integer.
b
)
b)
b
)
If
a
a
a
is odd then there do not exist positive integers
m
,
n
,
p
m, n, p
m
,
n
,
p
such that
m
n
p
mnp
mn
p
is even and
x
m
x
n
x
p
\frac{x_m} {x_nx_p}
x
n
x
p
x
m
is a perfect square.
number theory