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National and Regional Contests
China Contests
China Team Selection Test
1997 China Team Selection Test
3
China integral sequence built by three rules
China integral sequence built by three rules
Source: China TST 1997, problem 3
May 22, 2005
calculus
integration
algebra unsolved
algebra
Problem Statement
Prove that there exists
m
∈
N
m \in \mathbb{N}
m
∈
N
such that there exists an integral sequence
{
a
n
}
\lbrace a_n \rbrace
{
a
n
}
which satisfies: I.
a
0
=
1
,
a
1
=
337
a_0 = 1, a_1 = 337
a
0
=
1
,
a
1
=
337
; II.
(
a
n
+
1
a
n
−
1
−
a
n
2
)
+
3
4
(
a
n
+
1
+
a
n
−
1
−
2
a
n
)
=
m
,
∀
(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall
(
a
n
+
1
a
n
−
1
−
a
n
2
)
+
4
3
(
a
n
+
1
+
a
n
−
1
−
2
a
n
)
=
m
,
∀
n
≥
1
n \geq 1
n
≥
1
; III.
1
6
(
a
n
+
1
)
(
2
a
n
+
1
)
\frac{1}{6}(a_n + 1)(2a_n + 1)
6
1
(
a
n
+
1
)
(
2
a
n
+
1
)
is a perfect square
∀
\forall
∀
n
≥
1
n \geq 1
n
≥
1
.
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