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National and Regional Contests
Portugal Contests
Portugal MO
2019 Portugal MO
2019 Portugal MO
Part of
Portugal MO
Subcontests
(5)
6
1
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metro network with n stations
A metro network with
n
≥
2
n \ge 2
n
≥
2
stations, where each station is connected to each of the others by a one-way line, is said to be dispersed i f there are two stations
A
A
A
and
B
B
B
such that it is not possible to go from
A
A
A
to
B
B
B
through is from the network. If a network is dispersed, but it is possible to choose a station
A
A
A
and reverse the direction of all lines to and from
A
A
A
so that the new network is no longer dispersed, the network is said to be correctable. Indicates all integers
n
n
n
for which there is a network with
n
n
n
stations, dispersed and not correctable.
4
1
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2 colors in 2x3 grid
On a board with
3
3
3
columns and
4
4
4
rows, each of the
12
12
12
squares will be painted green or white. In the first and last row, the number of squares painted green must be the same. Furthermore, in the first and last column, the number of squares painted green must also be unequal. How many different ways can you paint the board?
2
1
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balanced 5-digit integers
A five-digit integer is said to be balanced i f the sum of any three of its digits is divisible by any of the other two. How many balanced numbers are there?
1
1
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area wanted, joining midpoints with vertices in square 2019 Portugal p1
In a square of side
10
10
10
cm , the vertices are joined to the midpoints on the opposite sides, as shown in the figure. How much does the area of the colored region measure? https://1.bp.blogspot.com/-bHrc1Nu0PQI/X4KaJysLAcI/AAAAAAAAMk0/LLGv1fotQO0Tk1AXqQymG_nNdpyWcbjyACLcBGAsYHQ/s109/2019%2BPortugal%2Bp1.png
5
1
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LQ = BQ wanted, equal arcs on circumcircle of a triangle
Let
[
A
B
C
]
[ABC]
[
A
BC
]
be a acute-angled triangle and its circumscribed circle
Γ
\Gamma
Γ
. Let
D
D
D
be the point on the line
A
B
AB
A
B
such that
A
A
A
is the midpoint of the segment
[
D
B
]
[DB]
[
D
B
]
and
P
P
P
is the point of intersection of
C
D
CD
C
D
with
Γ
\Gamma
Γ
. Points
W
W
W
and
L
L
L
lie on the smaller arcs \overarc{BC} and \overarc{AB}, respectively, and are such that \overarc{BW} = \overarc{LA }= \overarc{AP}. The
L
C
LC
L
C
and
A
W
AW
A
W
lines intersect at
Q
Q
Q
. Shows that
L
Q
=
B
Q
LQ = BQ
L
Q
=
BQ
.