MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2008 Singapore MO Open
2008 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(5)
5
1
Hide problems
SMO 2008 q5
consider a
2008
×
2008
2008 \times 2008
2008
×
2008
chess board. let
M
M
M
be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of
M
M
M
. (eg for
2
×
3
2\times 3
2
×
3
chessboard, the value of
M
M
M
is 3.)
4
1
Hide problems
SMO 2008 q4
let
0
<
a
,
b
<
π
/
2
0<a,b<\pi/2
0
<
a
,
b
<
π
/2
. Show that
5
c
o
s
2
(
a
)
+
5
s
i
n
2
(
a
)
s
i
n
2
(
b
)
c
o
s
2
(
b
)
≥
27
c
o
s
(
a
)
+
36
s
i
n
(
a
)
\frac{5}{cos^2(a)}+\frac{5}{sin^2(a)sin^2(b)cos^2(b)} \geq 27cos(a)+36sin(a)
co
s
2
(
a
)
5
+
s
i
n
2
(
a
)
s
i
n
2
(
b
)
co
s
2
(
b
)
5
≥
27
cos
(
a
)
+
36
s
in
(
a
)
3
1
Hide problems
SMO 2008 q3
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
2
1
Hide problems
SMO 2008 q2
in the acute triangle
△
A
B
C
\triangle ABC
△
A
BC
. M is a point in the interior of the segment AC and N is a point on the extension of segment AC such that MN=AC. let D,E be the feet of perpendiculars from M,N onto lines BC,AB respectively prove that the orthocentre of
△
A
B
C
\triangle ABC
△
A
BC
lies on circumcircle of
△
B
E
D
\triangle BED
△
BE
D
1
1
Hide problems
(n+1)^k-1=n!
Find all pairs of positive integers
(
n
,
k
)
(n,k)
(
n
,
k
)
so that (n\plus{}1)^k\minus{}1\equal{}n!.