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1999 Turkey MO (2nd round)
4
Turkey NMO 1999, P-4, find all sequences of real numbers
Turkey NMO 1999, P-4, find all sequences of real numbers
Source:
December 23, 2010
algebra proposed
algebra
Problem Statement
Find all sequences
a
1
,
a
2
,
.
.
.
,
a
2000
{{a}_{1}},{{a}_{2}},...,{{a}_{2000}}
a
1
,
a
2
,
...
,
a
2000
of real numbers such that
∑
n
=
1
2000
a
n
=
1999
\sum\limits_{n=1}^{2000}{{{a}_{n}}=1999}
n
=
1
∑
2000
a
n
=
1999
and such that
1
2
<
a
n
<
1
\frac{1}{2}<{{a}_{n}}<1
2
1
<
a
n
<
1
and
a
n
+
1
=
a
n
(
2
−
a
n
)
{{a}_{n+1}}={{a}_{n}}(2-{{a}_{n}})
a
n
+
1
=
a
n
(
2
−
a
n
)
for all
n
≥
1
n\ge 1
n
≥
1
.
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