MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2003 Singapore MO Open
2003 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(4)
1
1
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exist m,n such a_m = a_n and a_{m+1} = b_{n+1} in an English alphabet
A sequence
(
a
1
,
a
2
,
.
.
.
,
a
675
)
(a_1,a_2,...,a_{675})
(
a
1
,
a
2
,
...
,
a
675
)
is given so that each term is an alphabet in the English language (no distinction is made between lower and upper case letters). It is known that in the sequence
a
a
a
is never followed by
b
b
b
and
c
c
c
is never followed by
d
d
d
. Show that there are integers
m
m
m
and
n
n
n
with
1
≤
m
<
n
≤
674
1 \le m < n \le 674
1
≤
m
<
n
≤
674
such that
a
m
=
a
n
a_m = a_n
a
m
=
a
n
and
a
m
+
1
=
a
n
+
1
a_{m+1} = a_{n+1}
a
m
+
1
=
a
n
+
1
·
3
1
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x^2 + y^2 + p^z = 2003 diophantine
For any given prime
p
p
p
, determine whether the equation
x
2
+
y
2
+
p
z
=
2003
x^2 + y^2 + p^z = 2003
x
2
+
y
2
+
p
z
=
2003
always has integer solutions in
x
,
y
,
z
x, y, z
x
,
y
,
z
. Justify your answer
2
1
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max of (xyz)/(1 + 5x)(4x + 3y)(5y + 6z)(z + 18) when x,y,z>0
Find the maximum value of
x
y
z
(
1
+
5
x
)
(
4
x
+
3
y
)
(
5
y
+
6
z
)
(
z
+
18
)
\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}
(
1
+
5
x
)
(
4
x
+
3
y
)
(
5
y
+
6
z
)
(
z
+
18
)
x
yz
as
x
,
y
x, y
x
,
y
and
z
z
z
range over the set of all positive real numbers. Justify your answer.
4
1
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BS > CS, when pentagon ABCDE is the base of a pyramid with apex S
The pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
which is inscribed in a circle with
A
B
<
D
E
AB < DE
A
B
<
D
E
is the base of a pyramid with apex
S
S
S
. If the longest side from
S
S
S
is
S
A
SA
S
A
, prove that
B
S
>
C
S
BS > CS
BS
>
CS
.