MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
USAJMO
2024 USAJMO
3
3
Part of
2024 USAJMO
Problems
(1)
p^k divides term of sequence
Source: USAJMO 2024/3
3/20/2024
Let
a
(
n
)
a(n)
a
(
n
)
be the sequence defined by
a
(
1
)
=
2
a(1)=2
a
(
1
)
=
2
and
a
(
n
+
1
)
=
(
a
(
n
)
)
n
+
1
−
1
a(n+1)=(a(n))^{n+1}-1
a
(
n
+
1
)
=
(
a
(
n
)
)
n
+
1
−
1
for each integer
n
≥
1
n\geq 1
n
≥
1
. Suppose that
p
>
2
p>2
p
>
2
is a prime and
k
k
k
is a positive integer. Prove that some term of the sequence
a
(
n
)
a(n)
a
(
n
)
is divisible by
p
k
p^k
p
k
.Proposed by John Berman
AMC
USA(J)MO
USAJMO