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Contests
International Contests
IberoAmerican
2008 IberoAmerican
2008 IberoAmerican
Part of
IberoAmerican
Subcontests
(6)
6
1
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biribol tournament
Biribol is a game played between two teams of 4 people each (teams are not fixed). Find all the possible values of
n
n
n
for which it is possible to arrange a tournament with
n
n
n
players in such a way that every couple of people plays a match in opposite teams exactly once.
5
1
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Iberoamerican Olympiad 2008, problem 5
Let
A
B
C
ABC
A
BC
a triangle and
X
X
X
,
Y
Y
Y
and
Z
Z
Z
points at the segments
B
C
BC
BC
,
A
C
AC
A
C
and
A
B
AB
A
B
, respectively.Let
A
′
A'
A
′
,
B
′
B'
B
′
and
C
′
C'
C
′
the circuncenters of triangles
A
Z
Y
AZY
A
Z
Y
,
B
X
Z
BXZ
BXZ
,
C
Y
X
CYX
C
Y
X
, respectively.Prove that
4
(
A
′
B
′
C
′
)
≥
(
A
B
C
)
4(A'B'C')\geq(ABC)
4
(
A
′
B
′
C
′
)
≥
(
A
BC
)
with equality if and only if
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
are concurrents. Note:
(
X
Y
Z
)
(XYZ)
(
X
Y
Z
)
denotes the area of
X
Y
Z
XYZ
X
Y
Z
4
1
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Iberoamerican Olympiad 2008, problem 4
Prove that the equation x^{2008}\plus{} 2008!\equal{} 21^{y} doesn't have solutions in integers.
1
1
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maxima and minima in 2008x2008 board
The integers from 1 to
200
8
2
2008^2
200
8
2
are written on each square of a
2008
×
2008
2008 \times 2008
2008
×
2008
board. For every row and column the difference between the maximum and minimum numbers is computed. Let
S
S
S
be the sum of these 4016 numbers. Find the greatest possible value of
S
S
S
.
2
1
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lines concurrent with external bisector
Given a triangle
A
B
C
ABC
A
BC
, let
r
r
r
be the external bisector of
∠
A
B
C
\angle ABC
∠
A
BC
.
P
P
P
and
Q
Q
Q
are the feet of the perpendiculars from
A
A
A
and
C
C
C
to
r
r
r
. If CP \cap BA \equal{} M and AQ \cap BC\equal{}N, show that
M
N
MN
MN
,
r
r
r
and
A
C
AC
A
C
concur.
3
1
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P(x) - P(y) divisible by 107
Let P(x) \equal{} x^3 \plus{} mx \plus{} n be an integer polynomial satisfying that if P(x) \minus{} P(y) is divisible by 107, then x \minus{} y is divisible by 107 as well, where
x
x
x
and
y
y
y
are integers. Prove that 107 divides
m
m
m
.