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Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1952 Moscow Mathematical Olympiad
1952 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(26)
224-
1
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MMO 224- Moscow MO 1952 locus of vertices C of acute ABC
You are given a segment
A
B
AB
A
B
. Find the locus of the vertices
C
C
C
of acute-angled triangles
A
B
C
ABC
A
BC
.
230
1
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MMO 230 Moscow MO 1952 200 soldiers, 20x10, tallest, shortest, bet
200
200
200
soldiers occupy in a rectangle (military call it a square and educated military a carree):
20
20
20
men (per row) times
10
10
10
men (per column). In each row, we consider the tallest man (if some are of equal height, choose any of them) and of the
10
10
10
men considered we select the shortest (if some are of equal height, choose any of them). Call him
A
A
A
. Next the soldiers assume their initial positions and in each column the shortest soldier is selected, of these
20
20
20
, the tallest is chosen. Call him
B
B
B
. Two colonels bet on which of the two soldiers chosen by these two distinct procedures is taller:
A
A
A
or
B
B
B
. Which colonel wins the bet?
232
1
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MMO 232 Moscow MO 1952 2x^{2m}+ax^m+3 do not / 3x^{2n}+ax^n+2
Prove that for any integer
a
a
a
the polynomial
3
x
2
n
+
a
x
n
+
2
3x^{2n}+ax^n+2
3
x
2
n
+
a
x
n
+
2
cannot be divided by
2
x
2
m
+
a
x
m
+
3
2x^{2m}+ax^m+3
2
x
2
m
+
a
x
m
+
3
without a remainder.
231
1
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MMO 231 Moscow MO 1952 cos32x +a_{31}cos3x+... +a_2cos2x+a_1cosx
Prove that for arbitrary fixed
a
1
,
a
2
,
.
.
,
a
31
a_1, a_2,.. , a_{31}
a
1
,
a
2
,
..
,
a
31
the sum
cos
32
x
+
a
31
cos
31
x
+
.
.
.
+
a
2
c
o
s
2
x
+
a
1
cos
x
\cos 32x + a_{31} \cos 31x +... + a_2 cos 2x + a_1 \cos x
cos
32
x
+
a
31
cos
31
x
+
...
+
a
2
cos
2
x
+
a
1
cos
x
can take both positive and negative values as
x
x
x
varies.
229
1
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MMO 229 Moscow MO 1952 angle chasing in a triangle 80-80-20
In an isosceles triangle
△
A
B
C
,
∠
A
B
C
=
2
0
o
\vartriangle ABC, \angle ABC = 20^o
△
A
BC
,
∠
A
BC
=
2
0
o
and
B
C
=
A
B
BC = AB
BC
=
A
B
. Points
P
P
P
and
Q
Q
Q
are chosen on sides
B
C
BC
BC
and
A
B
AB
A
B
, respectively, so that
∠
P
A
C
=
5
0
o
\angle PAC = 50^o
∠
P
A
C
=
5
0
o
and
∠
Q
C
A
=
6
0
o
\angle QCA = 60^o
∠
QC
A
=
6
0
o
. Prove that
∠
P
Q
C
=
3
0
o
\angle PQC = 30^o
∠
PQC
=
3
0
o
.
228
1
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MMO 228 Moscow MO 1952 3 cylinders into a cube
How to arrange three right circular cylinders of diameter
a
/
2
a/2
a
/2
and height
a
a
a
into an empty cube with side
a
a
a
so that the cylinders could not change position inside the cube? Each cylinder can, however, rotate about its axis of symmetry.
227
1
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MMO 227 Moscow MO 1952 99 straight lines divide a plane into n parts
99
99
99
straight lines divide a plane into
n
n
n
parts. Find all possible values of
n
n
n
less than
199
199
199
.
226
1
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MMO 226 Moscow MO 1952 7-digit numbers from 1, 2, 3, 4, 5, 6, 7
Seven chips are numbered
1
,
2
,
3
,
4
,
5
,
6
,
7
1, 2, 3, 4, 5, 6, 7
1
,
2
,
3
,
4
,
5
,
6
,
7
. Prove that none of the seven-digit numbers formed by these chips is divisible by any other of these seven-digit numbers.
225
1
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MMO 225 Moscow MO 1952 length is mean proportional of 2 lenghts
From a point
C
C
C
, tangents
C
A
CA
C
A
and
C
B
CB
CB
are drawn to a circle
O
O
O
. From an arbitrary point
N
N
N
on the circle, perpendiculars
N
D
,
N
E
,
N
F
ND, NE, NF
N
D
,
NE
,
NF
are drawn on
A
B
,
C
A
AB, CA
A
B
,
C
A
and
C
B
CB
CB
, respectively. Prove that the length of
N
D
ND
N
D
is the mean proportional of the lengths of
N
E
NE
NE
and
N
F
NF
NF
.
224
1
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MMO 224 Moscow MO 1952 square of number begins with 0.9...9
a) Prove that if the square of a number begins with
0.
9...9
⏟
100
0.\underbrace{\hbox{9...9}}_{\hbox{100}}
0.
100
9...9
, then the number itself begins with
0.
9...9
⏟
100
0.\underbrace{\hbox{9...9}}_{\hbox{100}}
0.
100
9...9
,.b) Calculate
0.9...9
\sqrt{0.9...9}
0.9...9
(
60
60
60
nines) to
60
60
60
decimal places
223
1
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MMO 223 Moscow MO 1952 tangent incircles wanted when AB + CD = BC + AD
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, let
A
B
+
C
D
=
B
C
+
A
D
AB + CD = BC + AD
A
B
+
C
D
=
BC
+
A
D
. Prove that the circle inscribed in
A
B
C
ABC
A
BC
is tangent to the circle inscribed in
A
C
D
ACD
A
C
D
.
222
1
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MMO 222 Moscow MO 1952 15x15 system of equations (1 - x_1x_2 = 0)
a) Solve the system of equations
{
1
−
x
1
x
2
=
0
1
−
x
2
x
3
=
0
.
.
.
1
−
x
14
x
15
=
0
1
−
x
15
x
1
=
0
\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{14}x_{15} = 0 \\ 1 - x_{15}x_1 = 0 \end{cases}
⎩
⎨
⎧
1
−
x
1
x
2
=
0
1
−
x
2
x
3
=
0
...
1
−
x
14
x
15
=
0
1
−
x
15
x
1
=
0
b) Solve the system of equations
{
1
−
x
1
x
2
=
0
1
−
x
2
x
3
=
0
.
.
.
1
−
x
n
−
1
x
n
=
0
1
−
x
n
x
1
=
0
\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{n-1}x_{n} = 0 \\ 1 - x_{n}x_1 = 0 \end{cases}
⎩
⎨
⎧
1
−
x
1
x
2
=
0
1
−
x
2
x
3
=
0
...
1
−
x
n
−
1
x
n
=
0
1
−
x
n
x
1
=
0
How does the solution vary for distinct values of
n
n
n
?
221
1
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MMO 221 Moscow MO 1952 all roots real and positive ax^2+bx+c+p=0
Prove that if for any positive
p
p
p
all roots of the equation
a
x
2
+
b
x
+
c
+
p
=
0
ax^2 + bx + c + p = 0
a
x
2
+
b
x
+
c
+
p
=
0
are real and positive then
a
=
0
a = 0
a
=
0
.
220
1
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MMO 220 Moscow MO 1952 sphere inscribed in a trihedral angle
A sphere with center at
O
O
O
is inscribed in a trihedral angle with vertex
S
S
S
. Prove that the plane passing through the three tangent points is perpendicular to
O
S
OS
OS
.
219
1
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MMO 219 Moscow MO 1952 (1-x)^n + (1 + x)^n < 2^n if n>= 2, |x| < 1
Prove that
(
1
−
x
)
n
+
(
1
+
x
)
n
<
2
n
(1 - x)^n + (1 + x)^n < 2^n
(
1
−
x
)
n
+
(
1
+
x
)
n
<
2
n
for an integer
n
≥
2
n \ge 2
n
≥
2
and
∣
x
∣
<
1
|x| < 1
∣
x
∣
<
1
.
218
1
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MMO 218 Moscow MO 1952 arcsin(cos(arcsin x)) , arccos(sin(arccos x))
How
a
r
c
sin
(
cos
(
a
r
c
sin
x
)
)
arc \sin(\cos(arc \sin x))
a
rc
sin
(
cos
(
a
rc
sin
x
))
and
a
r
c
cos
(
sin
(
a
r
c
cos
x
)
)
arc \cos(\sin(arc \cos x))
a
rc
cos
(
sin
(
a
rc
cos
x
))
are related with each other?
217
1
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MMO 217 Moscow MO 1952 3 skew lines are pair-wise perpendicular to
Given three skew lines. Prove that they are pair-wise perpendicular to their pair-wise perpendiculars.
216
1
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MMO 216 Moscow MO 1952 integer sequence with sum of squares
A sequence of integers is constructed as follows:
a
1
a_1
a
1
is an arbitrary three-digit number,
a
2
a_2
a
2
is the sum of squares of the digits of
a
1
,
a
3
a_1, a_3
a
1
,
a
3
is the sum of squares of the digits of
a
2
a_2
a
2
, etc. Prove that either
1
1
1
or
4
4
4
must occur in the sequence
a
1
,
a
2
,
a
3
,
.
.
.
.
a_1, a_2, a_3, ....
a
1
,
a
2
,
a
3
,
....
215
1
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MMO 215 Moscow MO 1952 inequality with inradii
△
A
B
C
\vartriangle ABC
△
A
BC
is divided by a straight line
B
D
BD
B
D
into two triangles. Prove that the sum of the radii of circles inscribed in triangles
A
B
D
ABD
A
B
D
and
D
B
C
DBC
D
BC
is greater than the radius of the circle inscribed in
△
A
B
C
\vartriangle ABC
△
A
BC
.
214
1
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MMO 214 Moscow MO 1952 |x| < 1, |y| < 1 => |(x - y/(1-xy)| <1
Prove that if
∣
x
∣
<
1
|x| < 1
∣
x
∣
<
1
and
∣
y
∣
<
1
|y| < 1
∣
y
∣
<
1
, then
∣
x
−
y
1
−
x
y
∣
<
1
\left|\frac{x - y}{1 -xy}\right|< 1
1
−
x
y
x
−
y
<
1
.
213
1
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MMO 213 Moscow MO 1952 geometric progression, integer denominator
Given a geometric progression whose denominator
q
q
q
is an integer not equal to
0
0
0
or
−
1
-1
−
1
, prove that the sum of two or more terms in this progression cannot equal any other term in it.
212
1
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MMO 212 Moscow MO 1952 equilateral criterion, orthocenter related
Prove that if the orthocenter divides all heights of a triangle in the same proportion, the triangle is equilateral.
211
1
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MMO 211 Moscow MO 1952 two men and a girl, walk, bike, time, 15km distance
Two men,
A
A
A
and
B
B
B
, set out from town
M
M
M
to town
N
N
N
, which is
15
15
15
km away. Their walking speed is
6
6
6
km/hr. They also have a bicycle which they can ride at
15
15
15
km/hr. Both
A
A
A
and
B
B
B
start simultaneously,
A
A
A
walking and
B
B
B
riding a bicycle until
B
B
B
meets a pedestrian girl,
C
C
C
, going from
N
N
N
to
M
M
M
. Then
B
B
B
lends his bicycle to
C
C
C
and proceeds on foot;
C
C
C
rides the bicycle until she meets
A
A
A
and gives
A
A
A
the bicycle which
A
A
A
rides until he reaches
N
N
N
. The speed of
C
C
C
is the same as that of
A
A
A
and
B
B
B
. The time spent by
A
A
A
and
B
B
B
on their trip is measured from the moment they started from
M
M
M
until the arrival of the last of them at
N
N
N
.a) When should the girl
C
C
C
leave
N
N
N
for
A
A
A
and
B
B
B
to arrive simultaneously in
N
N
N
? b) When should
C
C
C
leave
N
N
N
to minimize this time?
210
1
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MMO 210 Moscow MO 1952 all faces of parallelepiped are equal # => rhombuses.
Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.
209
1
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MMO 209 Moscow MO 1952 (ax+by+cz)^2+(bx+cy+az)^2+(cx+ay+bz)^2=..
Prove the identity:a)
(
a
x
+
b
y
+
c
z
)
2
+
(
b
x
+
c
y
+
a
z
)
2
+
(
c
x
+
a
y
+
b
z
)
2
=
(
c
x
+
b
y
+
a
z
)
2
+
(
b
x
+
a
y
+
c
z
)
2
+
(
a
x
+
c
y
+
b
z
)
2
(ax + by + cz)^2 + (bx + cy + az)^2 + (cx + ay + bz)^2 =(cx + by + az)^2 + (bx + ay + cz)^2 + (ax + cy + bz)^2
(
a
x
+
b
y
+
cz
)
2
+
(
b
x
+
cy
+
a
z
)
2
+
(
c
x
+
a
y
+
b
z
)
2
=
(
c
x
+
b
y
+
a
z
)
2
+
(
b
x
+
a
y
+
cz
)
2
+
(
a
x
+
cy
+
b
z
)
2
b)
(
a
x
+
b
y
+
c
z
+
d
u
)
2
+
(
b
x
+
c
y
+
d
z
+
a
u
)
2
+
(
c
x
+
d
y
+
a
z
+
b
u
)
2
+
(
d
x
+
a
y
+
b
z
+
c
u
)
2
=
(ax + by + cz + du)^2+(bx + cy + dz + au)^2 +(cx + dy + az + bu)^2 + (dx + ay + bz + cu)^2 =
(
a
x
+
b
y
+
cz
+
d
u
)
2
+
(
b
x
+
cy
+
d
z
+
a
u
)
2
+
(
c
x
+
d
y
+
a
z
+
b
u
)
2
+
(
d
x
+
a
y
+
b
z
+
c
u
)
2
=
(
d
x
+
c
y
+
b
z
+
a
u
)
2
+
(
c
x
+
b
y
+
a
z
+
d
u
)
2
+
(
b
x
+
a
y
+
d
z
+
c
u
)
2
+
(
a
x
+
d
y
+
c
z
+
b
u
)
2
(dx + cy + bz + au)^2+(cx + by + az + du)^2 +(bx + ay + dz + cu)^2 + (ax + dy + cz + bu)^2
(
d
x
+
cy
+
b
z
+
a
u
)
2
+
(
c
x
+
b
y
+
a
z
+
d
u
)
2
+
(
b
x
+
a
y
+
d
z
+
c
u
)
2
+
(
a
x
+
d
y
+
cz
+
b
u
)
2
.
208
1
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MMO 208 Moscow MO 1952 angle of touchpoints with incircle is acute
The circle is inscribed in
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
L
,
M
,
N
L, M, N
L
,
M
,
N
be the tangent points of the circle with sides
A
B
,
A
C
,
B
C
AB, AC, BC
A
B
,
A
C
,
BC
, respectively. Prove that
∠
M
L
N
\angle MLN
∠
M
L
N
is always an acute angle.