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National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1937 Moscow Mathematical Olympiad
1937 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(6)
037
1
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MMO 037 Moscow MO 1937 diagonals cut a n-gon in parts
Into how many parts can a convex
n
n
n
-gon be divided by its diagonals if no three diagonals meet at one point?
036
1
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MMO 036 Moscow MO 1937 ways that planes cut regular dodecahedron
* Given a regular dodecahedron. Find how many ways are there to draw a plane through it so that its section of the dodecahedron is a regular hexagon?
035
1
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MMO 035 Moscow MO 1937 three circles costruction
Given three points that are not on the same straight line. Three circles pass through each pair of the points so that the tangents to the circles at their intersection points are perpendicular to each other. Construct the circles.
034
1
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MMO 034 Moscow MO 1937 volume of tetrahedron is constant
Two segments slide along two skew lines. On each straight line there is a segment. Consider the tetrahedron with vertices at the endpoints of the segments. Prove that the volume of the tetrahedron does not depend on the position of the segments
033
1
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MMO 033 Moscow MO 1937 point M on line so MA +MB equal given sum
* On a plane two points
A
A
A
and
B
B
B
are on the same side of a line. Find point
M
M
M
on the line such that
M
A
+
M
B
MA +MB
M
A
+
MB
is equal to a given length.
032
1
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MMO 032 Moscow MO 1937 x+y+z = a,x^2+y^2+z^2=a^2, x^3 +y^3+z^3=a^3
Solve the system
{
x
+
y
+
z
=
a
x
2
+
y
2
+
z
2
=
a
2
x
3
+
y
3
+
z
3
=
a
3
\begin{cases} x+ y +z = a \\ x^2 + y^2 + z^2 = a^2 \\ x^3 + y^3 +z^3 = a^3 \end{cases}
⎩
⎨
⎧
x
+
y
+
z
=
a
x
2
+
y
2
+
z
2
=
a
2
x
3
+
y
3
+
z
3
=
a
3