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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2014 Switzerland - Final Round
5
5
Part of
2014 Switzerland - Final Round
Problems
(1)
n | a_n if sum_{d | n} a_d = 2^n
Source: Switzerland - 2014 Swiss MO Final Round p5
12/30/2022
Let
a
1
,
a
2
,
.
.
.
a_1, a_2, ...
a
1
,
a
2
,
...
a sequence of integers such that for every
n
∈
N
n \in N
n
∈
N
we have:
∑
d
∣
n
a
d
=
2
n
.
\sum_{d | n} a_d = 2^n.
d
∣
n
∑
a
d
=
2
n
.
Show for every
n
∈
N
n \in N
n
∈
N
that
n
n
n
divides
a
n
a_n
a
n
.Remark: For
n
=
6
n = 6
n
=
6
the equation is
a
1
+
a
2
+
a
3
+
a
6
=
2
6
.
a_1 + a_2 + a_3 + a_6 = 2^6.
a
1
+
a
2
+
a
3
+
a
6
=
2
6
.
number theory
divides
divisible