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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Junior Math Olympiad
2023 Singapore Junior Math Olympiad
2023 Singapore Junior Math Olympiad
Part of
Singapore Junior Math Olympiad
Subcontests
(5)
4
1
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Prime numbers and average value
Two distinct 2-digit prime numbers
p
,
q
p,q
p
,
q
can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of
p
p
p
and
q
q
q
, find all possible pairs
{
p
,
q
}
\{p,q\}
{
p
,
q
}
.
3
1
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Sliding dominoes
Define a domino to be a
1
×
2
1\times 2
1
×
2
rectangular block. A
2023
×
2023
2023\times 2023
2023
×
2023
square grid is filled with non-overlapping dominoes, leaving a single
1
×
1
1\times 1
1
×
1
gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the
3
×
3
3\times 3
3
×
3
grid shown below, John could move 2 dominoes:
D
D
D
, followed by
A
A
A
.) [asy] unitsize(18); draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle); draw((0,1)--(3,1)); draw((2,0)--(2,3)); draw((1,1)--(1,3)); label("A",(0.5,2)); label("B",(1.5,2)); label("C",(2.5,2)); label("D",(1,0.5)); [/asy]
2
1
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Arranging integers in a circle with 3-digit product of adjacent terms
What is the maximum number of integers that can be chosen from
1
,
2
,
…
,
99
1,2,\dots,99
1
,
2
,
…
,
99
so that the chosen integers can be arranged in a circle with the property that the product of every pair of neighbouring integers is 3-digit number?
1
1
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Angles in a Quadrilateral
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, the diagonals intersect at
O
O
O
, and
M
M
M
and
N
N
N
are points on the segments
O
A
OA
O
A
and
O
D
OD
O
D
respectively. Suppose
M
N
MN
MN
is parallel to
A
D
AD
A
D
and
N
C
NC
NC
is parallel to
A
B
AB
A
B
. Prove that
∠
A
B
M
=
∠
N
C
D
\angle ABM=\angle NCD
∠
A
BM
=
∠
NC
D
.
5
1
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Diophantine Equation
Find all positive integers
k
k
k
such that there exists positive integers
a
,
b
a, b
a
,
b
such that
a
2
+
4
=
(
k
2
−
4
)
b
2
.
a^2+4=(k^2-4)b^2.
a
2
+
4
=
(
k
2
−
4
)
b
2
.