MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian APMO Camp Selection Test
2024 Malaysian APMO Camp Selection Test
1
1
Part of
2024 Malaysian APMO Camp Selection Test
Problems
(1)
Eventually double of the previous term
Source: Own. Malaysian APMO CST 2024 P1
2/24/2024
Let
a
1
<
a
2
<
⋯
a_1<a_2< \cdots
a
1
<
a
2
<
⋯
be a strictly increasing sequence of positive integers. Suppose there exist
N
N
N
such that for all
n
>
N
n>N
n
>
N
,
a
n
+
1
∣
a
1
+
a
2
+
⋯
+
a
n
a_{n+1}\mid a_1+a_2+\cdots+a_n
a
n
+
1
∣
a
1
+
a
2
+
⋯
+
a
n
Prove that there exist
M
M
M
such that
a
m
+
1
=
2
a
m
a_{m+1}=2a_m
a
m
+
1
=
2
a
m
for all
m
>
M
m>M
m
>
M
.Proposed by Ivan Chan Kai Chin
number theory