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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2013 Singapore MO Open
2013 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(5)
3
1
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smo open 2nd round 2013 q3
Let n be a positve integer. prove there exists a positive integer n st
n
2013
−
n
20
+
n
13
−
2013
n^{2013}-n^{20}+n^{13}-2013
n
2013
−
n
20
+
n
13
−
2013
has at least N distinct prime factors.
5
1
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Thrice and Angle and Integral Sides
Let
A
B
C
ABC
A
BC
be a triangle with integral side lengths such that
∠
A
=
3
∠
B
\angle A=3\angle B
∠
A
=
3∠
B
. Find the minimum value of its perimeter.
4
1
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Sum and Sets of Integers
Let
F
F
F
be a finite non-empty set of integers and let
n
n
n
be a positive integer. Suppose that
∙
\bullet
∙
Any
x
∈
F
x \in F
x
∈
F
may be written as
x
=
y
+
z
x=y+z
x
=
y
+
z
for some
y
y
y
,
z
∈
F
z \in F
z
∈
F
;
∙
\bullet
∙
If
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
and
x
1
x_1
x
1
, ...,
x
k
∈
F
x_k \in F
x
k
∈
F
, then
x
1
+
⋯
+
x
k
≠
0
x_1+\cdots+x_k \neq 0
x
1
+
⋯
+
x
k
=
0
. Show that
F
F
F
has at least
2
n
+
2
2n+2
2
n
+
2
elements.
2
1
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Medians and stuff regarding it
Let
A
B
C
ABC
A
BC
be an acute-angled triangle and let
D
D
D
,
E
E
E
, and
F
F
F
be the midpoints of
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
respectively. Construct a circle, centered at the orthocenter of triangle
A
B
C
ABC
A
BC
, such that triangle
A
B
C
ABC
A
BC
lies in the interior of the circle. Extend
E
F
EF
EF
to intersect the circle at
P
P
P
,
F
D
FD
F
D
to intersect the circle at
Q
Q
Q
and
D
E
DE
D
E
to intersect the circle at
R
R
R
. Show that
A
P
=
B
Q
=
C
R
AP=BQ=CR
A
P
=
BQ
=
CR
.
1
1
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2013th powered sequence
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
, ... be a sequence of integers defined recursively by
a
1
=
2013
a_1=2013
a
1
=
2013
and for
n
≥
1
n \ge 1
n
≥
1
,
a
n
+
1
a_{n+1}
a
n
+
1
is the sum of the
2013
2013
2013
-th powers of the digits of
a
n
a_n
a
n
. Do there exist distinct positive integers
i
i
i
,
j
j
j
such that
a
i
=
a
j
a_i=a_j
a
i
=
a
j
?