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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Junior Math Olympiad
2015 Singapore Junior Math Olympiad
2015 Singapore Junior Math Olympiad
Part of
Singapore Junior Math Olympiad
Subcontests
(4)
4
1
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any x,y in set determine a unique isosceles triangle
Let
A
A
A
be a set of numbers chosen from
1
,
2
,
.
.
.
,
2015
1,2,..., 2015
1
,
2
,
...
,
2015
with the property that any two distinct numbers, say
x
x
x
and
y
y
y
, in
A
A
A
determine a unique isosceles triangle (which is non equilateral) whose sides are of length
x
x
x
or
y
y
y
. What is the largest possible size of
A
A
A
?
5
1
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k^k +1 is divisible by 30
Find all positive integers
k
k
k
such that
k
k
+
1
k^k +1
k
k
+
1
is divisible by
30
30
30
. Justify your answer.
3
1
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30 children, seated clockwise in a circle on the floor, sequence
There are
30
30
30
children,
a
1
,
a
2
,
.
.
.
,
a
30
a_1,a_2,...,a_{30}
a
1
,
a
2
,
...
,
a
30
seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of
1000
1000
1000
candies. She drops a candy behind the first child
a
1
a_1
a
1
. She then skips one child and drops a candy behind the third child,
a
3
a_3
a
3
. Now she skips two children and drops a candy behind the next child,
a
6
a_6
a
6
. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.
2
1
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triangles with equal areas inside a convex hexagon with parallel sides
In a convex hexagon
A
B
C
D
E
F
,
A
B
ABCDEF, AB
A
BC
D
EF
,
A
B
is parallel to
D
E
,
B
C
DE, BC
D
E
,
BC
is parallel to
E
F
EF
EF
and
C
D
CD
C
D
is parallel to
F
A
FA
F
A
. Prove that the triangles
A
C
E
ACE
A
CE
and
B
D
F
BDF
B
D
F
have the same area.