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Problems
Contests
National and Regional Contests
Turkey Contests
Akdeniz University MO
1997 Akdeniz University MO
3
3
Part of
1997 Akdeniz University MO
Problems
(2)
sequence
Source:
1/30/2016
(
x
n
)
(x_n)
(
x
n
)
be a sequence with
x
1
=
0
x_1=0
x
1
=
0
,
x
n
+
1
=
5
x
n
+
24
x
n
2
+
1
x_{n+1}=5x_n + \sqrt{24x_n^2+1}
x
n
+
1
=
5
x
n
+
24
x
n
2
+
1
. Prove that for
k
≥
2
k \geq 2
k
≥
2
x
k
x_k
x
k
is a natural number.
equation
Sequence
Sequences
number theory
9 $\mid n$ :D
Source:
1/30/2016
Let for all
k
∈
N
k \in {\mathbb N}
k
∈
N
k
k
k
's sum of the digits is
T
(
k
)
T(k)
T
(
k
)
. If a natural number
n
n
n
such that
T
(
n
)
=
T
(
1997
n
)
T(n)=T(1997n)
T
(
n
)
=
T
(
1997
n
)
, prove that
9
∣
n
9\mid n
9
∣
n
number theory