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Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2010 Philippine MO
2010 Philippine MO
Part of
Philippine MO
Subcontests
(5)
5
1
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2010 Philippines MO P5
Determine, with proof, the smallest positive integer
n
n
n
with the following property: For every choice of
n
n
n
integers, there exist at least two whose sum or difference is divisible by
2009
2009
2009
.
4
1
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2010 Philippines MO P4
There are
2008
2008
2008
blue,
2009
2009
2009
red and
2010
2010
2010
yellow chips on a table. At each step, one chooses two chips of different colors, and recolor both of them using the third color. Can all the chips be of the same color after some steps? Prove your answer.
3
1
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2010 Philippines MO P3
Let
R
∗
\mathbb{R}^*
R
∗
be the set of all real numbers, except
1
1
1
. Find all functions
f
:
R
∗
→
R
f:\mathbb{R}^* \rightarrow \mathbb{R}
f
:
R
∗
→
R
that satisfy the functional equation
x
+
f
(
x
)
+
2
f
(
x
+
2009
x
−
1
)
=
2010
x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010
x
+
f
(
x
)
+
2
f
(
x
−
1
x
+
2009
)
=
2010
.
2
1
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2010 Philippines MO P2
On a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
, there is a point
P
P
P
on side
A
D
AD
A
D
such that the triangle
C
D
P
CDP
C
D
P
and the quadrilateral
A
B
C
P
ABCP
A
BCP
have equal perimeters and equal areas. Prove that two sides of
A
B
C
D
ABCD
A
BC
D
have equal lengths.
1
1
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2010 Philippines MO P1
Find all primes that can be written both as a sum of two primes and as a difference of two primes.