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Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1956 Moscow Mathematical Olympiad
1956 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(26)
345
1
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MMO 345 Moscow MO 1956 equal faces of a triangular pyramid critetion
* Prove that if the trihedral angles at each of the vertices of a triangular pyramid are formed by the identical planar angles, then all faces of this pyramid are equal.
344
1
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MMO 344 Moscow MO 1956 3 nodes of a graph paper, acute
* Let
A
,
B
,
C
A, B, C
A
,
B
,
C
be three nodes of a graph paper. Prove that if
△
A
B
C
\vartriangle ABC
△
A
BC
is an acute one, then there is at least one more node either inside
△
A
B
C
\vartriangle ABC
△
A
BC
or on one of its sides.
343
1
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MMO 343 Moscow MO 1956 tangential ABCD, concurrency or // of tangents
A quadrilateral is circumscribed around a circle. Prove that the straight lines connecting neighboring tangent points either meet on the extension of a diagonal of the quadrilateral or are parallel to it.
342
1
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MMO 342 Moscow MO 1956 sequences with absolute values
Given three numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
denote the absolute values of the differences of each pair by
x
1
,
y
1
,
z
1
x_1,y_1, z_1
x
1
,
y
1
,
z
1
. From
x
1
,
y
1
,
z
1
x_1, y_1, z_1
x
1
,
y
1
,
z
1
form in the same fashion the numbers
x
2
,
y
2
,
z
2
x_2, y_2, z_2
x
2
,
y
2
,
z
2
, etc. It is known that
x
n
=
x
,
y
n
=
y
,
z
n
=
z
x_n = x,y_n = y, z_n = z
x
n
=
x
,
y
n
=
y
,
z
n
=
z
for some
n
n
n
. Find
y
y
y
and
z
z
z
if
x
=
1
x = 1
x
=
1
.
341
1
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MMO 341 Moscow MO 1956 1956 points in a cube of edge 13
1956
1956
1956
points are chosen in a cube with edge
13
13
13
. Is it possible to fit inside the cube a cube with edge
1
1
1
that would not contain any of the selected points?
340
1
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MMO 340 Moscow MO 1956 rectangles inside rectange, area overlap
a) * In a rectangle of area
5
5
5
sq. units,
9
9
9
rectangles of area
1
1
1
are arranged. Prove that the area of the overlap of some two of these rectangles is
≥
1
/
9
\ge 1/9
≥
1/9
b) In a rectangle of area
5
5
5
sq. units, lie
9
9
9
arbitrary polygons each of area
1
1
1
. Prove that the area of the overlap of some two of these rectangles is
≥
1
/
9
\ge 1/9
≥
1/9
339
1
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MMO 339 Moscow MO 1956 projections of segment on concurrent lines
Find the union of all projections of a given line segment
A
B
AB
A
B
to all lines passing through a given point
O
O
O
.
338
1
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MMO 338 Moscow MO 1956 11 trucks of 1.5 tons may carry 13.5 tons
* A shipment of
13.5
13.5
13.5
tons is packed in a number of weightless containers. Each loaded container weighs not more than
350
350
350
kg. Prove that
11
11
11
trucks each of which is capable of carrying ·
1.5
1.5
1.5
ton can carry this load.
337
1
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MMO 337 Moscow MO 1956 15 / umber of tree’s leaves
* Assume that the number of a tree’s leaves is a multiple of
15
15
15
. Neglecting the shade of the trunk and branches prove that one can rip off the tree
7
/
15
7/15
7/15
of its leaves so that not less than
8
/
15
8/15
8/15
of its shade remains.
336
1
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MMO 336 Moscow MO 1956 64 numbers with 1956 sum, square table
64
64
64
non-negative numbers whose sum equals
1956
1956
1956
are arranged in a square table, eight numbers in each row and each column. The sum of the numbers on the two longest diagonals is equal to
112
112
112
. The numbers situated symmetrically with respect to any of the longest diagonals are equal. (a) Prove that the sum of numbers in any column is less than
1035
1035
1035
.(b) Prove that the sum of numbers in any row is less than
518
518
518
.
335
1
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MMO 335 Moscow MO 1956 100 numbers in a row, sum conditions, question
a)
100
100
100
numbers (some positive, some negative) are written in a row. All of the following three types of numbers are underlined: 1) every positive number, 2) every number whose sum with the number following it is positive, 3) every number whose sum with the two numbers following it is positive. Can the sum of all underlined numbers be (i) negative? (ii) equal to zero?b)
n
n
n
numbers (some positive and some negative) are written in a row. Each positive number and each number whose sum with several of the numbers following it is positive is underlined. Prove that the sum of all underlined numbers is positive.
334
1
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MMO 334 Moscow MO 1956 4 perpendiculars in regular hexagon
a) Points
A
1
,
A
2
,
A
3
,
A
4
,
A
5
,
A
6
A_1, A_2, A_3, A_4, A_5, A_6
A
1
,
A
2
,
A
3
,
A
4
,
A
5
,
A
6
divide a circle of radius
1
1
1
into six equal arcs. Ray
ℓ
1
\ell_1
ℓ
1
from
A
1
A_1
A
1
connects
A
1
A_1
A
1
with
A
2
A_2
A
2
, ray
ℓ
2
\ell_2
ℓ
2
from
A
2
A_2
A
2
connects
A
2
A_2
A
2
with
A
3
A_3
A
3
, and so on, ray
ℓ
6
\ell_6
ℓ
6
from
A
6
A_6
A
6
connects
A
6
A_6
A
6
with
A
1
A_1
A
1
. From a point
B
1
B_1
B
1
on
ℓ
1
\ell_1
ℓ
1
the perpendicular is drawn on
ℓ
6
\ell_6
ℓ
6
, from the foot of this perpendicular another perpendicular is drawn on
ℓ
5
\ell_5
ℓ
5
, and so on. Let the foot of the
6
6
6
-th perpendicular coincide with
B
1
B_1
B
1
. Find the length of segment
A
1
B
1
A_1B_1
A
1
B
1
.b) Find points
B
1
,
B
2
,
.
.
.
,
B
n
B_1, B_2,... , B_n
B
1
,
B
2
,
...
,
B
n
on the extensions of sides
A
1
A
2
,
A
2
A
3
,
.
.
.
,
A
n
A
1
A_1A_2, A_2A_3,... , A_nA_1
A
1
A
2
,
A
2
A
3
,
...
,
A
n
A
1
of a regular
n
n
n
-gon
A
1
A
2
.
.
.
A
n
A_1A_2...A_n
A
1
A
2
...
A
n
such that
B
1
B
2
⊥
A
1
A
2
B_1B_2 \perp A_1A_2
B
1
B
2
⊥
A
1
A
2
,
B
2
B
3
⊥
A
2
A
3
B_2B_3 \perp A_2A_3
B
2
B
3
⊥
A
2
A
3
,
.
.
.
. . .
...
,
B
n
B
1
⊥
A
n
A
1
B_nB_1 \perp A_nA_1
B
n
B
1
⊥
A
n
A
1
.
333
1
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MMO 333 Moscow MO 1956 altitudes concurrent in circumcenter
Let
O
O
O
be the center of the circle circumscribed around
△
A
B
C
\vartriangle ABC
△
A
BC
, let
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
be symmetric to
O
O
O
through respective sides of
△
A
B
C
\vartriangle ABC
△
A
BC
. Prove that all altitudes of
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
pass through
O
O
O
, and all altitudes of
△
A
B
C
\vartriangle ABC
△
A
BC
pass through the center of the circle circumscribed around
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
.
332
1
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MMO 332 Moscow MO 1956 4x4 parameter system
Prove that the system of equations
{
x
1
−
x
2
=
a
x
3
−
x
4
=
b
x
1
+
x
2
+
x
3
+
x
4
=
1
\begin{cases} x_1 - x_2 = a \\ x_3 - x_4 = b \\ x_1 + x_2 + x_3 + x_4 = 1\end{cases}
⎩
⎨
⎧
x
1
−
x
2
=
a
x
3
−
x
4
=
b
x
1
+
x
2
+
x
3
+
x
4
=
1
has at least one solution in positive numbers (
x
1
,
x
2
,
x
3
,
x
4
>
0
x_1 ,x_2 ,x_3 ,x_4>0
x
1
,
x
2
,
x
3
,
x
4
>
0
) if and only if
∣
a
∣
+
∣
b
∣
<
1
|a| + |b| < 1
∣
a
∣
+
∣
b
∣
<
1
.
331
1
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MMO 331 Moscow MO 1956 A_1B_1/B_1A_2 . A_2B_2/B_2A_3... A_nB_n/B_nA_1=1
Given a closed broken line
A
1
A
2
A
3
.
.
.
A
n
A_1A_2A_3...A_n
A
1
A
2
A
3
...
A
n
in space and a plane intersecting all its segments,
A
1
A
2
A_1A_2
A
1
A
2
at
B
1
,
A
2
A
3
B_1, A_2A_3
B
1
,
A
2
A
3
at
B
2
B_2
B
2
,
.
.
.
...
...
,
A
n
A
1
A_nA_1
A
n
A
1
at
B
n
B_n
B
n
, prove that
A
1
B
1
B
1
A
2
⋅
A
2
B
2
B
2
A
3
⋅
A
3
B
3
B
3
A
4
⋅
.
.
.
⋅
A
n
B
n
B
n
A
1
=
1
\frac{A_1B_1}{B_1A_2}\cdot \frac{A_2B_2}{B_2A_3}\cdot \frac{A_3B_3}{B_3A_4}\cdot ...\cdot \frac{A_nB_n}{B_nA_1}= 1
B
1
A
2
A
1
B
1
⋅
B
2
A
3
A
2
B
2
⋅
B
3
A
4
A
3
B
3
⋅
...
⋅
B
n
A
1
A
n
B
n
=
1
.
330
1
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MMO 330 Moscow MO 1956 r\sqrt2 < a < 2r, square inscribed in triangle
A square of side
a
a
a
is inscribed in a triangle so that two of the square’s vertices lie on the base, and the other two lie on the sides of the triangle. Prove that if
r
r
r
is the radius of the circle inscribed in the triangle, then
r
2
<
a
<
2
r
r\sqrt2 < a < 2r
r
2
<
a
<
2
r
.
329
1
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MMO 329 Moscow MO 1956 point on altitude of tetrahedron, given areas
Consider positive numbers
h
,
s
1
,
s
2
h, s_1, s_2
h
,
s
1
,
s
2
, and a spatial triangle
△
A
B
C
\vartriangle ABC
△
A
BC
. How many ways are there to select a point
D
D
D
so that the height of tetrahedron
A
B
C
D
ABCD
A
BC
D
drawn from
D
D
D
equals
h
h
h
, and the areas of faces
A
C
D
ACD
A
C
D
and
B
C
D
BCD
BC
D
equal
s
1
s_1
s
1
and
s
2
s_2
s
2
, respectively?
328
1
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MMO 328 Moscow MO 1956 concurrency of lines of midpoints of opposite sides
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, consider quadrilateral
K
L
M
N
KLMN
K
L
MN
formed by the centers of mass of triangles
A
B
C
,
B
C
D
,
D
B
A
,
C
D
A
ABC, BCD, DBA, CDA
A
BC
,
BC
D
,
D
B
A
,
C
D
A
. Prove that the straight lines connecting the midpoints of the opposite sides of quadrilateral
A
B
C
D
ABCD
A
BC
D
meet at the same point as the straight lines connecting the midpoints of the opposite sides of
K
L
M
N
KLMN
K
L
MN
.
327
1
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MMO 327 Moscow MO 1956 x =1/4 (a + b + c + d) infinife table
On an infinite sheet of graph paper a table is drawn so that in each square of the table stands a number equal to the arithmetic mean of the four adjacent numbers. Out of the table a piece is cut along the lines of the graph paper. Prove that the largest number on the piece always occurs at an edge, where
x
=
1
4
(
a
+
b
+
c
+
d
)
x = \frac14 (a + b + c + d)
x
=
4
1
(
a
+
b
+
c
+
d
)
.
326
1
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MMO 326 Moscow MO 1956 deleting after 2nd digit after decimal
a) In the decimal expression of a positive number,
a
a
a
, all decimals beginning with the third after the decimal point, are deleted (i.e., we take an approximation of
a
a
a
with accuracy to
0.01
0.01
0.01
with deficiency). The number obtained is divided by
a
a
a
and the quotient is similarly approximated with the same accuracy by a number
b
b
b
. What numbers
b
b
b
can be thus obtained? Write all their possible values. b) same as (a) but with accuracy to
0.001
0.001
0.001
c) same as (a) but with accuracy to
0.0001
0.0001
0.0001
325
1
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MMO 325 Moscow MO 1956 locus of midpoints, equal segments related
On sides
A
B
AB
A
B
and
C
B
CB
CB
of
△
A
B
C
\vartriangle ABC
△
A
BC
there are drawn equal segments,
A
D
AD
A
D
and
C
E
CE
CE
, respectively, of arbitrary length (but shorter than min(
A
B
,
B
C
AB,BC
A
B
,
BC
)). Find the locus of midpoints of all possible segments
D
E
DE
D
E
.
324
1
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MMO 324 Moscow MO 1956 points on circle of radius 1956, distances
a) What is the least number of points that can be chosen on a circle of length
1956
1956
1956
, so that for each of these points there is exactly one chosen point at distance
1
1
1
, and exactly one chosen point at distance
2
2
2
(distances are measured along the circle)?b) On a circle of length
15
15
15
there are selected
n
n
n
points such that for each of them there is exactly one selected point at distance
1
1
1
from it, and exactly one is selected point at distance
2
2
2
from it. (All distances are measured along the circle.) Prove that
n
n
n
is divisible by
10
10
10
.
323
1
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MMO 323 Moscow MO 1956 integer that divides (5m + 6), (8m + 7)
a) Find all integers that can divide both the numerator and denominator of the ratio
5
m
+
6
8
m
+
7
\frac{5m + 6}{8m + 7}
8
m
+
7
5
m
+
6
for an integer
m
m
m
.b) Let
a
,
b
,
c
,
d
,
m
a, b, c, d, m
a
,
b
,
c
,
d
,
m
be integers. Prove that if the numerator and denominator of the ratio
a
m
+
b
c
m
+
d
\frac{am + b}{cm+ d}
c
m
+
d
am
+
b
are both divisible by
k
k
k
, then so is
a
d
−
b
c
ad - bc
a
d
−
b
c
.
322
1
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MMO 322 Moscow MO 1956 even no of segments in broken line
A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.
321
1
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MMO 321 Moscow MO 1956 2-digit x , sum of digits same as 2x,3x,..,9x
Find all two-digit numbers
x
x
x
the sum of whose digits is the same as that of
2
x
2x
2
x
,
3
x
3x
3
x
, ... ,
9
x
9x
9
x
.
320
1
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MMO 320 Moscow MO 1956 triangles ABC,BCD,CDA,DAB not all acute
Prove that there are no four points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
on a plane such that all triangles
△
A
B
C
,
△
B
C
D
,
△
C
D
A
,
△
D
A
B
\vartriangle ABC,\vartriangle BCD, \vartriangle CDA, \vartriangle DAB
△
A
BC
,
△
BC
D
,
△
C
D
A
,
△
D
A
B
are acute ones.