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International Contests
IMO Longlists
1983 IMO Longlists
39
39
Part of
1983 IMO Longlists
Problems
(1)
Alpha is the root of the equation E(x) = x^3 - 5x -50 = 0
Source:
10/7/2010
If
α
\alpha
α
is the real root of the equation
E
(
x
)
=
x
3
−
5
x
−
50
=
0
E(x) = x^3 - 5x -50 = 0
E
(
x
)
=
x
3
−
5
x
−
50
=
0
such that
x
n
+
1
=
(
5
x
n
+
50
)
1
/
3
x_{n+1} = (5x_n + 50)^{1/3}
x
n
+
1
=
(
5
x
n
+
50
)
1/3
and
x
1
=
5
x_1 = 5
x
1
=
5
, where
n
n
n
is a positive integer, prove that:(a)
x
n
+
1
3
−
α
3
=
5
(
x
n
−
α
)
x_{n+1}^3 - \alpha^3 = 5(x_n - \alpha)
x
n
+
1
3
−
α
3
=
5
(
x
n
−
α
)
(b)
α
<
x
n
+
1
<
x
n
.
\alpha < x_{n+1} < x_n.
α
<
x
n
+
1
<
x
n
.
algebra unsolved
algebra