MathDB
Problems
Contests
International Contests
JBMO ShortLists
2022 JBMO Shortlist
N4
N4
Part of
2022 JBMO Shortlist
Problems
(1)
JBMO Shortlist 2022 N4
Source: JBMO Shortlist 2022
6/26/2023
Consider the sequence
u
0
,
u
1
,
u
2
,
.
.
.
u_0, u_1, u_2, ...
u
0
,
u
1
,
u
2
,
...
defined by
u
0
=
0
,
u
1
=
1
,
u_0 = 0, u_1 = 1,
u
0
=
0
,
u
1
=
1
,
and
u
n
=
6
u
n
−
1
+
7
u
n
−
2
u_n = 6u_{n - 1} + 7u_{n - 2}
u
n
=
6
u
n
−
1
+
7
u
n
−
2
for
n
≥
2
n \ge 2
n
≥
2
. Show that there are no non-negative integers
a
,
b
,
c
,
n
a, b, c, n
a
,
b
,
c
,
n
such that
a
b
(
a
+
b
)
(
a
2
+
a
b
+
b
2
)
=
c
2022
+
42
=
u
n
.
ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n.
ab
(
a
+
b
)
(
a
2
+
ab
+
b
2
)
=
c
2022
+
42
=
u
n
.
number theory
Sequence
recursive
Junior
Balkan
shortlist