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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2006 Singapore Senior Math Olympiad
2006 Singapore Senior Math Olympiad
Part of
Singapore Senior Math Olympiad
Subcontests
(5)
1
1
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a,a + d, a+ 2d are all prime numbers larger than 3, then d is multiple of 6
Let
a
,
d
a, d
a
,
d
be integers such that
a
,
a
+
d
,
a
+
2
d
a,a + d, a+ 2d
a
,
a
+
d
,
a
+
2
d
are all prime numbers larger than
3
3
3
. Prove that
d
d
d
is a multiple of
6
6
6
.
4
1
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tiling n triangular tiles to create convex equiangular hexagon v2
You have a large number of congruent equilateral triangular tiles on a table and you want to fit
n
n
n
of them together to make a convex equiangular hexagon (i.e. one whose interior angles are
12
0
o
120^o
12
0
o
) . Obviously,
n
n
n
cannot be any positive integer. The first three feasible
n
n
n
are
6
,
10
6, 10
6
,
10
and
13
13
13
. Determine if
19
19
19
and
20
20
20
are feasible .
3
1
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TP bisects < ATB, related to tangent circles
Two circles are tangent to each other internally at a point
T
T
T
. Let the chord
A
B
AB
A
B
of the larger circle be tangent to the smaller circle at a point
P
P
P
. Prove that the line TP bisects
∠
A
T
B
\angle ATB
∠
A
TB
.
2
1
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LA + MB = LM inside a cyclic ABCD, angle bisector and parallel related
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral, let the angle bisectors at
A
A
A
and
B
B
B
meet at
E
E
E
, and let the line through
E
E
E
parallel to side
C
D
CD
C
D
intersect
A
D
AD
A
D
at
L
L
L
and
B
C
BC
BC
at
M
M
M
. Prove that
L
A
+
M
B
=
L
M
LA + MB = LM
L
A
+
MB
=
L
M
.
5
1
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Persistent Numbers
In a non-recent edition of Ripley's Believe It or Not, it was stated that the number
N
=
526315789473684210
N = 526315789473684210
N
=
526315789473684210
is a persistent number, that is, if multiplied by any positive integer the resulting number always contains the ten digits
0
,
1
,
2
,
3
,
.
.
.
,
8
,
9
0, 1, 2, 3,..., 8, 9
0
,
1
,
2
,
3
,
...
,
8
,
9
in some order with possible repetitions. a) Prove or disprove the above statement. b) Are there any persistent numbers smaller than the above number?