MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2004 Singapore MO Open
2004 Singapore MO Open
Part of
Singapore Senior Math Olympiad
Subcontests
(4)
4
1
Hide problems
sum x_i/(1+(n-1)x_i) <=1 where 0 < ,x_i <= 1
If
0
<
x
1
,
x
2
,
.
.
.
,
x
n
≤
1
0 <x_1,x_2,...,x_n\le 1
0
<
x
1
,
x
2
,
...
,
x
n
≤
1
, where
n
≥
1
n \ge 1
n
≥
1
, show that
x
1
1
+
(
n
−
1
)
x
1
+
x
2
1
+
(
n
−
1
)
x
2
+
.
.
.
+
x
n
1
+
(
n
−
1
)
x
n
≤
1
\frac{x_1}{1+(n-1)x_1}+\frac{x_2}{1+(n-1)x_2}+...+\frac{x_n}{1+(n-1)x_n}\le 1
1
+
(
n
−
1
)
x
1
x
1
+
1
+
(
n
−
1
)
x
2
x
2
+
...
+
1
+
(
n
−
1
)
x
n
x
n
≤
1
1
1
Hide problems
k<= m(m-1)/n(n-1) for n-elements subsets of {1,...,m}
Let
m
,
n
m,n
m
,
n
be integers so that
m
≥
n
>
1
m \ge n > 1
m
≥
n
>
1
. Let
F
1
,
.
.
.
,
F
k
F_1,...,F_k
F
1
,
...
,
F
k
be a collection of
n
n
n
-element subsets of
{
1
,
.
.
.
,
m
}
\{1,...,m\}
{
1
,
...
,
m
}
so that
F
i
∩
F
j
F_i\cap F_j
F
i
∩
F
j
contains at most
1
1
1
element,
1
≤
i
<
j
≤
k
1 \le i < j \le k
1
≤
i
<
j
≤
k
. Show that
k
≤
m
(
m
−
1
)
n
(
n
−
1
)
k\le \frac{m(m-1)}{n(n-1)}
k
≤
n
(
n
−
1
)
m
(
m
−
1
)
2
1
Hide problems
x^2 + ax + b = 167 y, 1<=a,b<=2004 , no of ordered solutions (a,b)
Find the number of ordered pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of integers, where
1
≤
a
,
b
≤
2004
1 \le a, b \le 2004
1
≤
a
,
b
≤
2004
, such that
x
2
+
a
x
+
b
=
167
y
x^2 + ax + b = 167 y
x
2
+
a
x
+
b
=
167
y
has integer solutions in
x
x
x
and
y
y
y
. Justify your answer.
3
1
Hide problems
concyclic wanted, OD _|_ BC, intersecting circles related
Let
A
D
AD
A
D
be the common chord of two circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
. A line through
D
D
D
intersects
Γ
1
\Gamma_1
Γ
1
at
B
B
B
and
Γ
2
\Gamma_2
Γ
2
at
C
C
C
. Let
E
E
E
be a point on the segment
A
D
AD
A
D
, different from
A
A
A
and
D
D
D
. The line
C
E
CE
CE
intersect
Γ
1
\Gamma_1
Γ
1
at
P
P
P
and
Q
Q
Q
. The line
B
E
BE
BE
intersects
Γ
2
\Gamma_2
Γ
2
at
M
M
M
and
N
N
N
. (i) Prove that
P
,
Q
,
M
,
N
P,Q,M,N
P
,
Q
,
M
,
N
lie on the circumference of a circle
Γ
3
\Gamma_3
Γ
3
. (ii) If the centre of
Γ
3
\Gamma_3
Γ
3
is
O
O
O
, prove that
O
D
OD
O
D
is perpendicular to
B
C
BC
BC
.