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Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2007 Singapore MO Open
2007 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(2)
4
1
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SMO 2007 open q4
find all functions
f
:
N
→
N
f:\mathbb{N}\rightarrow\mathbb{N}
f
:
N
→
N
st
f
(
f
(
m
)
+
f
(
n
)
)
=
m
+
n
∀
m
,
n
∈
N
f(f(m)+f(n))=m+n \,\forall m,n\in\mathbb{N}
f
(
f
(
m
)
+
f
(
n
))
=
m
+
n
∀
m
,
n
∈
N
related: https://artofproblemsolving.com/community/c6h381298
3
1
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Isosceles triangle
Let
A
1
A_1
A
1
,
B
1
B_1
B
1
be two points on the base
A
B
AB
A
B
of an isosceles triangle
A
B
C
ABC
A
BC
, with
∠
C
>
6
0
∘
\angle C>60^{\circ}
∠
C
>
6
0
∘
, such that
∠
A
1
C
B
1
=
∠
A
B
C
\angle A_1CB_1=\angle ABC
∠
A
1
C
B
1
=
∠
A
BC
. A circle externally tangent to the circumcircle of
△
A
1
B
1
C
\triangle A_1B_1C
△
A
1
B
1
C
is tangent to the rays
C
A
CA
C
A
and
C
B
CB
CB
at points
A
2
A_2
A
2
and
B
2
B_2
B
2
, respectively. Prove that
A
2
B
2
=
2
A
B
A_2B_2=2AB
A
2
B
2
=
2
A
B
.