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Problems
Contests
National and Regional Contests
Vietnam Contests
Pre - Vietnam Mathematical Olympiad
2012 Pre - Vietnam Mathematical Olympiad
2
Determine the first term of a sequence
Determine the first term of a sequence
Source: Pre-VMO 2012 - Round 2 - Problem 6
December 26, 2011
algebra
polynomial
algebra proposed
Problem Statement
Let
(
a
n
)
(a_n)
(
a
n
)
defined by:
a
0
=
1
,
a
1
=
p
,
a
2
=
p
(
p
−
1
)
a_0=1, \; a_1=p, \; a_2=p(p-1)
a
0
=
1
,
a
1
=
p
,
a
2
=
p
(
p
−
1
)
,
a
n
+
3
=
p
a
n
+
2
−
p
a
n
+
1
+
a
n
,
∀
n
∈
N
a_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}
a
n
+
3
=
p
a
n
+
2
−
p
a
n
+
1
+
a
n
,
∀
n
∈
N
. Knowing that(i)
a
n
>
0
,
∀
n
∈
N
a_n>0, \; \forall n \in \mathbb{N}
a
n
>
0
,
∀
n
∈
N
.(ii)
a
m
a
n
>
a
m
+
1
a
n
−
1
,
∀
m
≥
n
≥
0
a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0
a
m
a
n
>
a
m
+
1
a
n
−
1
,
∀
m
≥
n
≥
0
.Prove that
∣
p
−
1
∣
≥
2
|p-1| \ge 2
∣
p
−
1∣
≥
2
.
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