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National and Regional Contests
Poland Contests
Polish MO Finals
1997 Polish MO Finals
1
1
Part of
1997 Polish MO Finals
Problems
(2)
find x_7
Source:
11/11/2005
The positive integers
x
1
,
x
2
,
.
.
.
,
x
7
x_1, x_2, ... , x_7
x
1
,
x
2
,
...
,
x
7
satisfy
x
6
=
144
x_6 = 144
x
6
=
144
,
x
n
+
3
=
x
n
+
2
(
x
n
+
1
+
x
n
)
x_{n+3} = x_{n+2}(x_{n+1}+x_n)
x
n
+
3
=
x
n
+
2
(
x
n
+
1
+
x
n
)
for
n
=
1
,
2
,
3
,
4
n = 1, 2, 3, 4
n
=
1
,
2
,
3
,
4
. Find
x
7
x_7
x
7
.
number theory unsolved
number theory
sequence and a_n=0
Source:
11/11/2005
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
.
induction
number theory unsolved
number theory