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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Junior Math Olympiad
2012 Singapore Junior Math Olympiad
2012 Singapore Junior Math Olympiad
Part of
Singapore Junior Math Olympiad
Subcontests
(5)
5
1
Hide problems
set of 15 coprime numbers out of 2011 contains a prime number
Suppose
S
=
{
a
1
,
a
2
,
.
.
.
,
a
15
}
S = \{a_1, a_2,..., a_{15}\}
S
=
{
a
1
,
a
2
,
...
,
a
15
}
is a set of
15
1 5
15
distinct positive integers chosen from
2
,
3
,
.
.
.
,
2012
2 , 3, ... , 2012
2
,
3
,
...
,
2012
such that every two of them are coprime. Prove that
S
S
S
contains a prime number. (Note: Two positive integers
m
,
n
m, n
m
,
n
are coprime if their only common factor is 1)
4
1
Hide problems
x_1 + x_2 +...+ x_n = 16, 1/x_1 + 1/x_2 +...+ 1/x_n =1
Determine the values of the positive integer
n
n
n
for which the following system of equations has a solution in positive integers
x
1
,
x
2
,
.
.
.
,
,
x
n
x_1, x_2,...,, x_n
x
1
,
x
2
,
...
,,
x
n
. Find all solutions for each such
n
n
n
.
{
x
1
+
x
2
+
.
.
.
+
x
n
=
16
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
=
1
\begin{cases} x_1 + x_2 +...+ x_n = 16 \\ \\ \dfrac{1}{x_1} + \dfrac{1}{x_2} +...+ \dfrac{1}{x_n} = 1\end{cases}
⎩
⎨
⎧
x
1
+
x
2
+
...
+
x
n
=
16
x
1
1
+
x
2
1
+
...
+
x
n
1
=
1
2
1
Hide problems
all digits exactly once in one of numbers A, A^2, A^ 3
Does there exist an integer
A
A
A
such that each of the ten digits
0
,
1
,
.
.
.
,
9
0, 1, . . . , 9
0
,
1
,
...
,
9
appears exactly once as a digit in exactly one of the numbers
A
,
A
2
,
A
3
A, A^2, A^ 3
A
,
A
2
,
A
3
?
1
1
Hide problems
collinear with the parallelogram's center (Singapore Junior 2012)
Let
O
O
O
be the centre of a parallelogram
A
B
C
D
ABCD
A
BC
D
and
P
P
P
be any point in the plane. Let
M
,
N
M, N
M
,
N
be the midpoints of
A
P
,
B
P
AP, BP
A
P
,
BP
, respectively and
Q
Q
Q
be the intersection of
M
C
MC
MC
and
N
D
ND
N
D
. Prove that
O
,
P
O, P
O
,
P
and
Q
Q
Q
are collinear.
3
1
Hide problems
circumcenter, excenter and vertex collinear (Singapore Junior 2012)
In
△
A
B
C
\vartriangle ABC
△
A
BC
, the external bisectors of
∠
A
\angle A
∠
A
and
∠
B
\angle B
∠
B
meet at a point
D
D
D
. Prove that the circumcentre of
△
A
B
D
\vartriangle ABD
△
A
B
D
and the points
C
,
D
C, D
C
,
D
lie on the same straight line.