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National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
2008 Saint Petersburg Mathematical Olympiad
4
St Petersburgh 2008 #4
St Petersburgh 2008 #4
Source:
July 23, 2011
induction
inequalities proposed
inequalities
Problem Statement
The numbers
x
1
,
.
.
.
x
100
x_1,...x_{100}
x
1
,
...
x
100
are written on a board so that
x
1
=
1
2
x_1=\frac{1}{2}
x
1
=
2
1
and for every
n
n
n
from
1
1
1
to
99
99
99
,
x
n
+
1
=
1
−
x
1
x
2
x
3
∗
.
.
.
∗
x
100
x_{n+1}=1-x_1x_2x_3*...*x_{100}
x
n
+
1
=
1
−
x
1
x
2
x
3
∗
...
∗
x
100
. Prove that
x
100
>
0.99
x_{100}>0.99
x
100
>
0.99
.
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