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National and Regional Contests
Portugal Contests
Portugal MO
2007 Portugal MO
2007 Portugal MO
Part of
Portugal MO
Subcontests
(6)
4
1
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decoratin a square blanket with a ribbon and buttons, 2007th button ?
Fernanda decided to decorate a square blanket with a ribbon and buttons, placing a button in the center of each square where the ribbon passes and forming the design indicated in the figure. If Fernanda sews the first button in the shaded square on line
0
0
0
, on which line does she sew the
2007
2007
2007
th button? https://cdn.artofproblemsolving.com/attachments/2/9/0c9c85ec6448ee3f6f363c8f4bcdd5209f53f6.png
5
1
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there are houses of at least five different colors.
Rua do Antonio has
100
100
100
houses numbered from
1
1
1
to
100
100
100
. Any house numbered with the difference between the numbers of two houses of the same color is a different color. Prove that on Rua do Antonio there are houses of at least five different colors.
3
1
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max n, multiple of all positive integers less than \sqrt{n}.
Determines the largest integer
n
n
n
that is a multiple of all positive integers less than
n
\sqrt{n}
n
.
1
1
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necklace with 20 pearls, blue and white
Joao had blue, white and red pearls and with them he made a necklace with
20
20
20
pearls that has as many blue as white pearls. João noticed that, regardless of how he cut the necklace into two parts, both with an even number of pearls, one of the parts would always have more blue pearls than white ones. How many red pearls are in Joao's necklace?
6
1
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M/m >= \sqrt3 for 6 houses with min /max distances 2007 Portugal p6
In a village, the maximum distance between two houses is
M
M
M
and the minimum distance is
m
m
m
. Prove that if the village has
6
6
6
houses, then
M
m
≥
3
\frac{M}{m} \ge \sqrt3
m
M
≥
3
.
2
1
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concurrent circumcircles wanted 2007 Portugal p2
Let
[
A
B
C
]
[ABC]
[
A
BC
]
be a triangle and
X
,
Y
X, Y
X
,
Y
and
Z
Z
Z
points on the sides
[
A
B
]
,
[
B
C
]
[AB], [BC]
[
A
B
]
,
[
BC
]
and
[
A
C
]
[AC]
[
A
C
]
, respectively. Prove that circumcircles of triangles
A
X
Z
,
B
X
Y
AXZ, BXY
A
XZ
,
BX
Y
and
C
Y
Z
CYZ
C
Y
Z
intersect at a point.