MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2008 Singapore MO Open
3
3
Part of
2008 Singapore MO Open
Problems
(1)
SMO 2008 q3
Source: SMO 2008, open, second round
3/30/2018
let n,m be positive integers st
m
>
n
≥
5
m>n\geq 5
m
>
n
≥
5
with m depending on n. consider the sequence
a
1
,
a
2
,
.
.
.
a
m
a_1,a_2,...a_m
a
1
,
a
2
,
...
a
m
where
a
i
=
i
a_i=i
a
i
=
i
for
i
=
1
,
.
.
.
,
n
i=1,...,n
i
=
1
,
...
,
n
a
n
+
j
=
a
3
j
+
a
3
j
−
1
+
a
3
j
−
2
a_{n+j}=a_{3j}+a_{3j-1}+a_{3j-2}
a
n
+
j
=
a
3
j
+
a
3
j
−
1
+
a
3
j
−
2
for
j
=
1
,
.
.
,
m
−
n
j=1,..,m-n
j
=
1
,
..
,
m
−
n
with
m
−
3
(
m
−
n
)
=
m-3(m-n)=
m
−
3
(
m
−
n
)
=
1 or 2, ie
a
m
=
a
m
−
k
+
a
m
−
k
−
1
+
a
m
−
k
−
2
a_m=a_{m-k}+a_{m-k-1}+a_{m-k-2}
a
m
=
a
m
−
k
+
a
m
−
k
−
1
+
a
m
−
k
−
2
where k=1 or 2 (Thus if
n
=
5
n=5
n
=
5
, the sequence is 1,2,3,4,5,6,15 and if
n
=
8
n=8
n
=
8
, the sequence is 1,2,3,4,5,6,7,8,6,15,21) Find
S
=
a
1
+
.
.
.
+
a
m
S=a_1+...+a_m
S
=
a
1
+
...
+
a
m
if (i)
n
=
2007
n=2007
n
=
2007
(ii)
n
=
2008
n=2008
n
=
2008
Sequence