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Contests
International Contests
May Olympiad
May Olympiad L1 - geometry
May Olympiad L1 - geometry
Part of
May Olympiad
Subcontests
(2)
1995.5
1
Hide problems
tortoise and lizard walk around a rectangle
A tortoise walks
60
60
60
meters per hour and a lizard walks at
240
240
240
meters per hour. There is a rectangle
A
B
C
D
ABCD
A
BC
D
where
A
B
=
60
AB =60
A
B
=
60
and
A
D
=
120
AD =120
A
D
=
120
. Both start from the vertex
A
A
A
and in the same direction (
A
→
B
→
D
→
A
A \to B \to D \to A
A
→
B
→
D
→
A
), crossing the edge of the rectangle. The lizard has the habit of advancing two consecutive sides of the rectangle, turning to go back one, turning to go forward two, turning to go back one and so on. How many times and in what places do the tortoise and the lizard meet when the tortoise completes its third turn?
1995.4
1
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smallest perimeter of length on surface on a triangular base
We have four white equilateral triangles of
3
3
3
cm on each side and join them by their sides to obtain a triangular base pyramid. At each edge of the pyramid we mark two red dots that divide it into three equal parts. Number the red dots, so that when you scroll them in the order they were numbered, result a path with the smallest possible perimeter. How much does that path measure?