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Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2011 Philippine MO
2011 Philippine MO
Part of
Philippine MO
Subcontests
(5)
5
1
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Chromatic Number Range
The chromatic number
χ
\chi
χ
of an (infinite) plane is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart. Prove that
4
≤
χ
≤
7
4 \leq \chi \leq 7
4
≤
χ
≤
7
.
4
1
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A Functional Equation
Find all (if there is one) functions
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
such that for all
x
∈
R
x\in\mathbb{R}
x
∈
R
,
f
(
f
(
x
)
)
+
x
f
(
x
)
=
1.
f(f(x))+xf(x)=1.
f
(
f
(
x
))
+
x
f
(
x
)
=
1.
3
1
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Consecutive Numbers Containing 2011 Primes
The
2011
2011
2011
th prime number is
17483
17483
17483
and the next prime is
17489
17489
17489
. Does there exist a sequence of
201
1
2011
2011^{2011}
201
1
2011
consecutive positive integers that contain exactly
2011
2011
2011
prime numbers?
2
1
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Perpendicularity from Midpoints
In triangle
A
B
C
ABC
A
BC
, let
X
X
X
and
Y
Y
Y
be the midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. On segment
B
C
BC
BC
, there is a point
D
D
D
, different from its midpoint, such that
∠
X
D
Y
=
∠
B
A
C
\angle{XDY}=\angle{BAC}
∠
X
D
Y
=
∠
B
A
C
. Prove that
A
D
⊥
B
C
AD\perp BC
A
D
⊥
BC
.
1
1
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Finite Set Property
Find all nonempty finite sets
X
X
X
of real numbers such that for all
x
∈
X
x\in X
x
∈
X
,
x
+
∣
x
∣
∈
X
x+|x| \in X
x
+
∣
x
∣
∈
X
.