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Vietnam Team Selection Test
2006 Vietnam Team Selection Test
3
Problem 6 vietnamese tst 2006
Problem 6 vietnamese tst 2006
Source: Vietnamese TST 2006
April 18, 2006
induction
number theory proposed
number theory
Problem Statement
The real sequence
{
a
n
∣
n
=
0
,
1
,
2
,
3
,
.
.
.
}
\{a_n|n=0,1,2,3,...\}
{
a
n
∣
n
=
0
,
1
,
2
,
3
,
...
}
defined
a
0
=
1
a_0=1
a
0
=
1
and
a
n
+
1
=
1
2
(
a
n
+
1
3
⋅
a
n
)
.
a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ).
a
n
+
1
=
2
1
(
a
n
+
3
⋅
a
n
1
)
.
Denote
A
n
=
3
3
⋅
a
n
2
−
1
.
A_n=\frac{3}{3 \cdot a_n^2-1}.
A
n
=
3
⋅
a
n
2
−
1
3
.
Prove that
A
n
A_n
A
n
is a perfect square and it has at least
n
n
n
distinct prime divisors.
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