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2007 Moldova National Olympiad
11.1
11.1
Part of
2007 Moldova National Olympiad
Problems
(1)
Moldova NMO, 2007, XI Grade, Problem 1
Source: Well known idea, however...
3/3/2007
Define the sequence
(
x
n
)
(x_{n})
(
x
n
)
:
x
1
=
1
3
x_{1}=\frac{1}{3}
x
1
=
3
1
and
x
n
+
1
=
x
n
2
+
x
n
x_{n+1}=x_{n}^{2}+x_{n}
x
n
+
1
=
x
n
2
+
x
n
. Find
[
1
x
1
+
1
+
1
x
2
+
1
+
⋯
+
1
x
2007
+
1
]
\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]
[
x
1
+
1
1
+
x
2
+
1
1
+
⋯
+
x
2007
+
1
1
]
, wehere
[
[
[
]
]
]
denotes the integer part.
algebra unsolved
algebra