MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1971 Dutch Mathematical Olympiad
4
4
Part of
1971 Dutch Mathematical Olympiad
Problems
(1)
S(k)=a(1) + a(2) + ... + a(2^k) when n = 2^{a(n)} (2b(n)+1),
Source: Netherlands - Dutch NMO 1971 p4
1/28/2023
For every positive integer
n
n
n
there exist unambiguously determined non-negative integers
a
(
n
)
a(n)
a
(
n
)
and
b
(
n
)
b(n)
b
(
n
)
such that
n
=
2
a
(
n
)
(
2
b
(
n
)
+
1
)
,
n = 2^{a(n)}(2b(n)+1),
n
=
2
a
(
n
)
(
2
b
(
n
)
+
1
)
,
For positive integer
k
k
k
we define
S
(
k
)
S(k)
S
(
k
)
by:
a
(
1
)
+
a
(
2
)
+
.
.
.
+
a
(
2
k
)
=
S
(
k
)
a(1) + a(2) + ... + a(2^k) = S(k)
a
(
1
)
+
a
(
2
)
+
...
+
a
(
2
k
)
=
S
(
k
)
Express
S
(
k
)
S(k)
S
(
k
)
in terms of
k
k
k
.
number theory
Sum