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Poland Contests
Polish MO Finals
1997 Polish MO Finals
1
sequence and a_n=0
sequence and a_n=0
Source:
November 11, 2005
induction
number theory unsolved
number theory
Problem Statement
The sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
is defined by
a
1
=
0
a_1 = 0
a
1
=
0
,
a
n
=
a
[
n
/
2
]
+
(
−
1
)
n
(
n
+
1
)
/
2
a_n = a_{[n/2]} + (-1)^{n(n+1)/2}
a
n
=
a
[
n
/2
]
+
(
−
1
)
n
(
n
+
1
)
/2
. Show that for any positive integer
k
k
k
we can find
n
n
n
in the range
2
k
≤
n
<
2
k
+
1
2^k \leq n < 2^{k+1}
2
k
≤
n
<
2
k
+
1
such that
a
n
=
0
a_n = 0
a
n
=
0
.
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