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Portugal MO
1997 Portugal MO
1997 Portugal MO
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Portugal MO
Subcontests
(6)
6
1
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n parallel segments of different (?) lengths
n
n
n
parallel segments of lengths
a
1
≤
a
2
≤
a
3
≤
.
.
.
≤
a
n
a_1 \le a_2 \le a_3 \le ... \le a_n
a
1
≤
a
2
≤
a
3
≤
...
≤
a
n
were painted to mark an airport atrium. However, the architect decided that the
n
n
n
segments should have equal length. If the cost per meter of extending the lines is equal to the cost of reducing them, how long should the lines be in order to minimize costs?
4
1
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2-way movements of dodo animal in coordinates
The dodo was a strange animal. As it has already become extinct, only conjectures can be made about its way of life. One of the most unique conjectures is linked to the way the dodo moved. It seems that an adult animal only moved by jumping, which could be of two types: type I:
1
1
1
meter to the East and
3
3
3
to the North; type II:
2
2
2
meters to the West and
4
4
4
to the South. a) Prove that it was possible for the diode to reach a point located
19
19
19
meters to the East and
95
95
95
to the North of it and determines the number of jumps for each type he needed to carry out. b) Prove that it was impossible for the diode to reach a point located
18
18
18
meters to the East and
95
95
95
meters to the North of it.
3
1
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20 cities and 2 airline companies
In Abaliba country there are twenty cities and two airline companies, Blue Planes and Red Planes. The flights are planned as follows:
∙
\bullet
∙
Given any two cities, one and only one of the two companies operates direct flights (in both directions and without stops) between the two cities. Furthermore:
∙
\bullet
∙
There are two cities A and B between which it is not possible to fly (with possible stops) using only Red Planes. Prove that, given any two cities, a passenger can travel from one to the other using only Blue Planes, making at most one stop in a third city.
1
1
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20 questions, 7-2-0 points, 87 total - Portugal OPM 1997 p1
A test has twenty questions. Seven points are awarded for each correct answer, two points are deducted for each incorrect answer and no points are awarded or deducted for each unanswered question. Joana obtained
87
87
87
points. How many questions did she not answer?
2
1
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isosceles wanted, midpoints, cube 1997 Portugal p2
Consider the cube
A
B
C
D
E
F
G
H
ABCDEFGH
A
BC
D
EFG
H
and denote by, respectively,
M
M
M
and
N
N
N
the midpoints of
[
A
B
]
[AB]
[
A
B
]
and
[
C
D
]
[CD]
[
C
D
]
. Let
P
P
P
be a point on the line defined by
[
A
E
]
[AE]
[
A
E
]
and
Q
Q
Q
the point of intersection of the lines defined by
[
P
M
]
[PM]
[
PM
]
and
[
B
F
]
[BF]
[
BF
]
. Prove that the triangle
[
P
Q
N
]
[PQN]
[
PQN
]
is isosceles. https://cdn.artofproblemsolving.com/attachments/0/0/57559efbad87903d087c738df279b055b4aefd.png
5
1
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total sum of line segments >70 in a square of side 12, 1997 Portugal p5
A square region of side
12
12
12
contains a water source that supplies an irrigation system constituted by several straight channels forming polygonal lines. Considers the source as a point and each channel as a line segment. Knowing that a point is irrigated if it is not more than
1
1
1
distance from any channel and that the system was designed so that the entire region is irrigated, proves that the total length of irrigation channels exceeds
70
70
70
.