MathDB
Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2008 Philippine MO
2008 Philippine MO
Part of
Philippine MO
Subcontests
(4)
4
1
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Function Summation Identity
Let
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
be a function defined by
f
(
x
)
=
200
8
2
x
2008
+
200
8
2
x
f(x)=\frac{2008^{2x}}{2008+2008^{2x}}
f
(
x
)
=
2008
+
200
8
2
x
200
8
2
x
. Prove that
f
(
1
2007
)
+
f
(
2
2007
)
+
⋯
+
f
(
2005
2007
)
+
f
(
2006
2007
)
=
1003.
\begin{aligned} f\left(\frac{1}{2007}\right)+f\left(\frac{2}{2007}\right)+\cdots+f\left(\frac{2005}{2007}\right)+f\left(\frac{2006}{2007}\right)=1003. \end{aligned}
f
(
2007
1
)
+
f
(
2007
2
)
+
⋯
+
f
(
2007
2005
)
+
f
(
2007
2006
)
=
1003.
3
1
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Midpoint Created by Tangents
Let
P
P
P
be a point outside a circle
Γ
\Gamma
Γ
, and let the two tangent lines through
P
P
P
touch
Γ
\Gamma
Γ
at
A
A
A
and
B
B
B
. Let
C
C
C
be on the minor arc
A
B
AB
A
B
, and let ray
P
C
PC
PC
intersect
Γ
\Gamma
Γ
again at
D
D
D
. Let
ℓ
\ell
ℓ
be the line through
B
B
B
and parallel to
P
A
PA
P
A
.
ℓ
\ell
ℓ
intersects
A
C
AC
A
C
and
A
D
AD
A
D
at
E
E
E
and
F
F
F
, respectively. Prove that
B
B
B
is the midpoint of
E
F
EF
EF
.
2
1
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Largest Integer and Divisibility
Find the largest integer
n
n
n
for which
n
2007
+
n
2006
+
⋯
+
n
2
+
n
+
1
n
+
2007
\frac{n^{2007}+n^{2006}+\cdots+n^2+n+1}{n+2007}
n
+
2007
n
2007
+
n
2006
+
⋯
+
n
2
+
n
+
1
is an integer.
1
1
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Disjoint Subsets with Same Sum
Prove that the set
{
1
,
2
,
⋯
,
2007
}
\{1, 2, \cdots, 2007\}
{
1
,
2
,
⋯
,
2007
}
can be expressed as the union of disjoint subsets
A
i
A_i
A
i
for
i
=
1
,
2
,
⋯
,
223
i=1,2,\cdots, 223
i
=
1
,
2
,
⋯
,
223
such that each
A
i
A_i
A
i
contains nine elements and the sum of all the elements in each
A
i
A_i
A
i
is the same.