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National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1938 Moscow Mathematical Olympiad
1938 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(5)
042
1
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MMO 042 Moscow MO 1938 no integers < 1000, not divided by 5 and 7
How many positive integers smaller than
1000
1000
1000
and not divisible by
5
5
5
and by
7
7
7
are there?
041
1
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MMO 041 Moscow MO 1938 triangle construction
Given the base, height and the difference between the angles at the base of a triangle, construct the triangle.
040
1
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MMO 040 Moscow MO 1938 max no of parts n planes divide space
What is the largest number of parts into which
n
n
n
planes can divide space?We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).
039
1
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MMO 039 Moscow MO 1938 reflections of points in space
The following operation is performed over points
O
1
,
O
2
,
O
3
O_1, O_2, O_3
O
1
,
O
2
,
O
3
and
A
A
A
in space. The point
A
A
A
is reflected with respect to
O
1
O_1
O
1
, the resultant point
A
1
A_1
A
1
is reflected through
O
2
O_2
O
2
, and the resultant point
A
2
A_2
A
2
through
O
3
O_3
O
3
. We get some point
A
3
A_3
A
3
that we will also consecutively reflect through
O
1
,
O
2
,
O
3
O_1, O_2, O_3
O
1
,
O
2
,
O
3
. Prove that the point obtained last coincides with
A
A
A
..
038
1
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MMO 038 Moscow MO 1938 equidistant planes from 4 points
In space
4
4
4
points are given. How many planes equidistant from these points are there? Consider separately (a) the generic case (the points given do not lie on a single plane) and (b) the degenerate cases.