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Contests
National and Regional Contests
Singapore Contests
Singapore Junior Math Olympiad
2019 Singapore Junior Math Olympiad
2019 Singapore Junior Math Olympiad
Part of
Singapore Junior Math Olympiad
Subcontests
(5)
4
1
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sqrt2 a^3+ 3/(ab-b^2) >= 10 when a>b>0
Let
a
>
b
>
0
a>b>0
a
>
b
>
0
. Prove that
2
a
3
+
3
a
b
−
b
2
≥
10
\sqrt2 a^3+ \frac{3}{ab-b^2}\ge 10
2
a
3
+
ab
−
b
2
3
≥
10
When does equality hold?
3
1
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(2m-1)/n and (2n-1)/m are both integers
Find all positive integers
m
,
n
m, n
m
,
n
such that
2
m
−
1
n
\frac{2m-1}{n}
n
2
m
−
1
and
2
n
−
1
m
\frac{2n-1}{m}
m
2
n
−
1
are both integers.
2
1
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315 marbles divided into three piles of 81, 115 and 119
There are
315
315
315
marbles divided into three piles of
81
,
115
81, 115
81
,
115
and
119
119
119
. In each move Ah Meng can either merge several piles into a single pile or divide a pile with an even number of marbles into
2
2
2
equal piles. Can Ah Meng divide the marbles into
315
315
315
piles, each with a single marble?
1
1
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AE=2EB inside a right isosceles triangle (Singapore Junior 2019)
In the triangle
A
B
C
,
A
C
=
B
C
,
∠
C
=
9
0
o
,
D
ABC, AC=BC, \angle C=90^o, D
A
BC
,
A
C
=
BC
,
∠
C
=
9
0
o
,
D
is the midpoint of
B
C
,
E
BC, E
BC
,
E
is the point on
A
B
AB
A
B
such that
A
D
AD
A
D
is perpendicular to
C
E
CE
CE
. Prove that
A
E
=
2
E
B
AE=2EB
A
E
=
2
EB
.
5
1
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Swapping Red and Blue Blocks
Let
n
n
n
be a positive integer and consider an arrangement of
2
n
2n
2
n
blocks in a straight line, where
n
n
n
of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let
A
A
A
be the minimum number of swaps needed to make the first
n
n
n
blocks all red and
B
B
B
be the minimum number of swaps needed to make the first
n
n
n
blocks all blue. Show that
A
+
B
A+B
A
+
B
is independent of the starting arrangement and determine its value.