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National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1935 Moscow Mathematical Olympiad
1935 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(21)
021
1
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MMO 021 Moscow MO 1935 lcm , gcd of 2 and 3 numbers
Denote by
M
(
a
,
b
,
c
,
.
.
.
,
k
)
M(a, b, c, . . . , k)
M
(
a
,
b
,
c
,
...
,
k
)
the least common multiple and by
D
(
a
,
b
,
c
,
.
.
.
,
k
)
D(a, b, c, . . . , k)
D
(
a
,
b
,
c
,
...
,
k
)
the greatest common divisor of
a
,
b
,
c
,
.
.
.
,
k
a, b, c, . . . , k
a
,
b
,
c
,
...
,
k
. Prove that:a)
M
(
a
,
b
)
D
(
a
,
b
)
=
a
b
M(a, b)D(a, b) = ab
M
(
a
,
b
)
D
(
a
,
b
)
=
ab
,b)
M
(
a
,
b
,
c
)
D
(
a
,
b
)
D
(
b
,
c
)
D
(
a
,
c
)
D
(
a
,
b
,
c
)
=
a
b
c
\frac{M(a, b, c)D(a, b)D(b, c)D(a, c)}{D(a, b, c)}= abc
D
(
a
,
b
,
c
)
M
(
a
,
b
,
c
)
D
(
a
,
b
)
D
(
b
,
c
)
D
(
a
,
c
)
=
ab
c
.
020
1
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MMO 020 Moscow MO 1935 n as a sum of 3 pos. integers
How many ways are there of representing a positive integer
n
n
n
as the sum of three positive integers? Representations which differ only in the order of the summands are considered to be distinct.
019
1
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MMO 019 Moscow MO 1935 coloring faces of cube / dodecahedron
a) How many distinct ways are there are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.)b)* How many distinct ways are there are there of painting the faces of a dodecahedron
12
12
12
different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.)
018
1
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MMO 018 Moscow MO 1935 1^3 + 3^3 + 5^3 +... + (2n - 1)^3
Evaluate the sum:
1
3
+
3
3
+
5
3
+
.
.
.
+
(
2
n
−
1
)
3
1^3 + 3^3 + 5^3 +... + (2n - 1)^3
1
3
+
3
3
+
5
3
+
...
+
(
2
n
−
1
)
3
.
017
1
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MMO 017 Moscow MO 1935 system x^3 - y^3 = 2b , x^2y - xy^2 = b
Solve the system
{
x
3
−
y
3
=
26
x
2
y
−
x
y
2
=
6
\begin{cases} x^3 - y^3 = 26 \\ x^2y - xy^2 = 6 \end{cases}
{
x
3
−
y
3
=
26
x
2
y
−
x
y
2
=
6
in
C
C
C
[hide=other version]solved below Solve the system
{
x
3
−
y
3
=
2
b
x
2
y
−
x
y
2
=
b
\begin{cases} x^3 - y^3 = 2b \\ x^2y - xy^2 = b \end{cases}
{
x
3
−
y
3
=
2
b
x
2
y
−
x
y
2
=
b
016
1
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MMO 016 Moscow MO 1935 system x+y=2 , xy - z^2 = 1
How many real solutions does the following system have ?
{
x
+
y
=
2
x
y
−
z
2
=
1
\begin{cases} x+y=2 \\ xy - z^2 = 1 \end{cases}
{
x
+
y
=
2
x
y
−
z
2
=
1
015
1
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MMO 015 Moscow MO 1935 concurrency in space
Triangles
△
A
B
C
\vartriangle ABC
△
A
BC
and
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
lie on different planes. Line
A
B
AB
A
B
intersects line
A
1
B
1
A_1B_1
A
1
B
1
, line
B
C
BC
BC
intersects line
B
1
C
1
B_1C_1
B
1
C
1
and line
C
A
CA
C
A
intersects line
C
1
A
1
C_1A_1
C
1
A
1
. Prove that either the three lines
A
A
1
,
B
B
1
,
C
C
1
AA_1, BB_1, CC_1
A
A
1
,
B
B
1
,
C
C
1
meet at one point or that they are all parallel.
014
1
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MMO 014 Moscow MO 1935 locus in space
Find the locus of points on the surface of a cube that serve as the vertex of the smallest angle that subtends the diagonal.
013
1
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MMO 013 Moscow MO 1935 triangle construction
The median, bisector, and height, all originate at the same vertex of a triangle. Given the intersection points of the median, bisector, and height with the circumscribed circle, construct the triangle.
012
1
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MMO 012 Moscow MO 1935 unfolding lateral surface of cone, angle wanted
The unfolding of the lateral surface of a cone is a sector of angle
12
0
o
120^o
12
0
o
. The angles at the base of a pyramid constitute an arithmetic progression with a difference of
1
5
o
15^o
1
5
o
. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle
α
\alpha
α
between the plane of this face and the base.
011
1
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MMO 011 Moscow MO 1935 sum of circumradii = another circumradius
In
△
A
B
C
\vartriangle ABC
△
A
BC
, two straight lines drawn from an arbitrary point
D
D
D
on
A
B
AB
A
B
are parallel to
A
C
AC
A
C
,
B
C
BC
BC
and intersect
B
C
BC
BC
,
A
C
AC
A
C
at
F
F
F
,
G
G
G
, respectively. Prove that the sum of the circumferences of the circles circumscribed around
△
A
D
G
\vartriangle ADG
△
A
D
G
and
△
B
D
F
\vartriangle BDF
△
B
D
F
is equal to the circumference of the circle circumscribed around
△
A
B
C
\vartriangle ABC
△
A
BC
.
010
1
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MMO 010 Moscow MO 1935 3 variable parameter system of equations
Solve the system
{
x
2
+
y
2
−
2
z
2
=
2
a
2
x
+
y
+
2
z
=
4
(
a
2
+
1
)
z
2
−
x
y
=
a
2
\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\ x + y + 2z = 4(a^2 + 1) \\ z^2 - xy = a^2 \end{cases}
⎩
⎨
⎧
x
2
+
y
2
−
2
z
2
=
2
a
2
x
+
y
+
2
z
=
4
(
a
2
+
1
)
z
2
−
x
y
=
a
2
009
1
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MMO 009 Moscow MO 1935 truncated cone, angle wanted
The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle
ϕ
\phi
ϕ
between the cone’s generating line and its base.
008
1
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MMO 008 Moscow MO 1935 sidelengths in arithmetic progression
Prove that if the lengths of the sides of a triangle form an arithmetic progression, then the radius of the inscribed circle is one third of one of the heights of the triangle.
007
1
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MMO 007 Moscow MO 1935 system of arithmetic + geometric progression
Find four consecutive terms
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
of an arithmetic progression and four consecutive terms
a
1
,
b
1
,
c
1
,
d
1
a_1, b_1, c_1, d_1
a
1
,
b
1
,
c
1
,
d
1
of a geometric progression such that
{
a
+
a
1
=
27
b
+
b
1
=
27
c
+
c
1
=
39
d
+
d
1
=
87
\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}
⎩
⎨
⎧
a
+
a
1
=
27
b
+
b
1
=
27
c
+
c
1
=
39
d
+
d
1
=
87
.
006
1
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MMO 006 Moscow MO 1935 volume of a pyramid, angle related
The base of a right pyramid is a quadrilateral whose sides are each of length
a
a
a
. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.
005
1
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MMO 005 Moscow MO 1935 square construction , 3 parallel lines related
Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.
004
1
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MMO 004 Moscow MO 1935 train, bridge, length, speed
A train passes an observer in
t
1
t_1
t
1
sec. At the same speed the train crosses a bridge
ℓ
\ell
ℓ
m long. It takes the train
t
2
t_2
t
2
sec to cross the bridge from the moment the locomotive drives onto the bridge until the last car leaves it. Find the length and speed of the train.
003
1
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MMO 003 Moscow MO 1935 angles in a pyramid
The base of a pyramid is an isosceles triangle with the vertex angle
α
\alpha
α
. The pyramid’s lateral edges are at angle
ϕ
\phi
ϕ
to the base. Find the dihedral angle
θ
\theta
θ
at the edge connecting the pyramid’s vertex to that of angle
α
\alpha
α
.
002
1
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MMO 002 Moscow MO 1935 triangle construction
Given the lengths of two sides of a triangle and that of the bisector of the angle between these sides, construct the triangle.
001
1
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MMO 001 Moscow MO 1935 ratio of means 25/24
Find the ratio of two numbers if the ratio of their arithmetic mean to their geometric mean is
25
:
24
25 : 24
25
:
24