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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
1996 Singapore MO Open
1996 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(4)
4
1
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x^3 + y^3 + z^3 = wx^2y^2z^2 diophantine
Determine all the solutions of the equation
x
3
+
y
3
+
z
3
=
w
x
2
y
2
z
2
x^3 + y^3 + z^3 = wx^2y^2z^2
x
3
+
y
3
+
z
3
=
w
x
2
y
2
z
2
in natural numbers
x
,
y
,
z
,
w
x, y, z, w
x
,
y
,
z
,
w
. Justify your answer
3
1
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(x + 2)^n - x^n = 1 + 7^n diophantine
Let
n
n
n
be a positive integer. Prove that there is no positive integer solution to thxe equation
(
x
+
2
)
n
−
x
n
=
1
+
7
n
(x + 2)^n - x^n = 1 + 7^n
(
x
+
2
)
n
−
x
n
=
1
+
7
n
.
2
1
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< PCQ in a unit square ABCD when AP+AQ+PQ = 2
In the following figure,
A
B
C
D
ABCD
A
BC
D
is a square of unit length and
P
,
Q
P, Q
P
,
Q
are points on
A
D
AD
A
D
and
A
B
AB
A
B
respectively. Find
∠
P
C
Q
\angle PCQ
∠
PCQ
if
∣
A
P
∣
+
∣
A
Q
∣
+
∣
P
Q
∣
=
2
|AP| + |AQ| + |PQ| = 2
∣
A
P
∣
+
∣
A
Q
∣
+
∣
PQ
∣
=
2
. https://cdn.artofproblemsolving.com/attachments/2/c/2f40db978c1d3fcbc0161f874b5cbec926058e.png
1
1
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probability 3 numbers in [0,1] are sidelengths
Three numbers are selected at random from the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
. What is the probability that they form the lengths of the sides of a triangle?