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Problems
Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1947 Moscow Mathematical Olympiad
1947 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(19)
135-
1
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MMO 135- Moscow MO 1947 4 points on plane, different distances
Position the
4
4
4
points on plane so that when measuring of all pairwise distances between them, it turned out only two different numbers. Find all such locations.
140
1
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MMO 140 Moscow MO 1947 4 faces of tetrahedron have same area => equal
Prove that if the four faces of a tetrahedron are of the same area they are equal.
139
1
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MMO 139 Moscow MO 1947 like pascal triangle
In the numerical triangle
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1..............
................1..............
................1..............
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1...1...1.........
...........1 ...1 ...1.........
...........1...1...1.........
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1...2...3...2...1....
......1... 2... 3 ... 2 ... 1....
......1...2...3...2...1....
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1...3...6...7...6...3...1
.1...3...6...7...6...3...1
.1...3...6...7...6...3...1
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...............................
...............................
each number is equal to the sum of the three nearest to it numbers from the row above it; if the number is at the beginning or at the end of a row then it is equal to the sum of its two nearest numbers or just to the nearest number above it (the lacking numbers above the given one are assumed to be zeros). Prove that each row, starting with the third one, contains an even number.
138
1
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MMO 138 Moscow MO 1947 n wire triangles in space
In space,
n
n
n
wire triangles are situated so that any two of them have a common vertex and each vertex is the vertex of
k
k
k
triangles. Find all
n
n
n
and
k
k
k
for which this is possible.
137
1
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MMO 137 Moscow MO 1947 101 no from set 1, ...,200, pair divides another
a)
101
101
101
numbers are selected from the set
1
,
2
,
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,
200
1, 2, . . . , 200
1
,
2
,
...
,
200
. Prove that among the numbers selected there is a pair in which one number is divisible by the other.b) One number less than
16
16
16
, and
99
99
99
other numbers are selected from the set
1
,
2
,
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.
.
,
200
1, 2, . . . , 200
1
,
2
,
...
,
200
. Prove that among the selected numbers there are two such that one divides the other.
136
1
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MMO 136 Moscow MO 1947 no convex 13-gon can be cut into #
Prove that no convex
13
13
13
-gon can be cut into parallelograms.
135
1
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MMO 135 Moscow MO 1947 4 out of 5 create a convex quadrilateral
a) Given
5
5
5
points on a plane, no three of which lie on one line. Prove that four of these points can be taken as vertices of a convex quadrilateral.b) Inside a square, consider a convex quadrilateral and inside the quadrilateral, take a point
A
A
A
. It so happens that no three of the
9
9
9
points — the vertices of the square, of the quadrilateral and
A
A
A
— lie on one line. Prove that
5
5
5
of these points are vertices of a convex pentagon.
134
1
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MMO 134 Moscow MO 1947 digits in decimal expression 2^{100}
How many digits are there in the decimal expression of
2
100
2^{100}
2
100
?
133
1
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MMO 133 Moscow MO 1947 20 cubes, two materials, <=11 weighings
Twenty cubes of the same size and appearance are made of either aluminum or of heavier duralumin. How can one find the number of duralumin cubes using not more than
11
11
11
weighings on a balance without weights? (We assume that all cubes can be made of aluminum, but not all of duralumin.)
132
1
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MMO 132 Moscow MO 1947 locus in space, equidistant
Given line
A
B
AB
A
B
and point
M
M
M
. Find all lines in space passing through
M
M
M
at distance
d
d
d
.
131
1
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MMO 131 Moscow MO 1947 product (1-1/10)(1-1/10^2)...(1-1/10^99)
Calculate (without calculators, tables, etc.) with accuracy to
0.00001
0.00001
0.00001
the product
(
1
−
1
10
)
(
1
−
1
1
0
2
)
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.
(
1
−
1
1
0
99
)
\left(1-\frac{1}{10}\right)\left(1-\frac{1}{10^2}\right)...\left(1-\frac{1}{10^{99}}\right)
(
1
−
10
1
)
(
1
−
1
0
2
1
)
...
(
1
−
1
0
99
1
)
130
1
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MMO 130 Moscow MO 1947 coeffiient of x^{20} in (1+x^2 -x^3)^{1000}
Which of the polynomials,
(
1
+
x
2
−
x
3
)
1000
(1+x^2 -x^3)^{1000}
(
1
+
x
2
−
x
3
)
1000
or
(
1
−
x
2
+
x
3
)
1000
(1-x^2 +x^3)^{1000}
(
1
−
x
2
+
x
3
)
1000
, has the greater coefficient of
x
20
x^{20}
x
20
after expansion and collecting the terms?
129
1
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MMO 129 Moscow MO 1947 squares in a 8x8 chessboard
How many squares different in size or location can be drawn on an
8
×
8
8 \times 8
8
×
8
chess board? Each square drawn must consist of whole chess board’s squares.
128
1
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MMO 128 Moscow MO 1947 coefficient (...(((x - 2)^2 - 2)^2 -2)^2-...-2)^2-2)^2
Find the coefficient of
x
2
x^2
x
2
after expansion and collecting the terms of the following expression (there are
k
k
k
pairs of parentheses):
(
(
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(
(
(
x
−
2
)
2
−
2
)
2
−
2
)
2
−
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−
2
)
2
−
2
)
2
((... (((x - 2)^2 - 2)^2 -2)^2 -... -2)^2 - 2)^2
((
...
(((
x
−
2
)
2
−
2
)
2
−
2
)
2
−
...
−
2
)
2
−
2
)
2
127
1
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MMO 127 Moscow MO 1947 3 equal circumcircles
Point
O
O
O
is the intersection point of the heights of an acute triangle
△
A
B
C
\vartriangle ABC
△
A
BC
. Prove that the three circles which pass: a) through
O
,
A
,
B
O, A, B
O
,
A
,
B
, b) through
O
,
B
,
C
O, B, C
O
,
B
,
C
, and c) through
O
,
C
,
A
O, C, A
O
,
C
,
A
, are equal
126
1
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MMO 126 Moscow MO 1947 lines from a point inside a convex pentagon
Given a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
, prove that if an arbitrary point
M
M
M
inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass.Note: In reality, we need to exclude the points of the diagonals, because that in this case the drawn lines can pass not through the internal points of the sides, but through the vertices. But if the drawn diagonals are not considered or counted twice (because they are drawn from two vertices), then the statement remains true.
125
1
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MMO 125 Moscow MO 1947 coefficients of x^{17} and x^{18} from (1+x^5+x^7)^{20}
Find the coefficients of
x
17
x^{17}
x
17
and
x
18
x^{18}
x
18
after expansion and collecting the terms of
(
1
+
x
5
+
x
7
)
20
(1+x^5+x^7)^{20}
(
1
+
x
5
+
x
7
)
20
.
124
1
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MMO 124 Moscow MO 1947 one relative prime in 4 of 5 consecutive
a) Prove that of
5
5
5
consecutive positive integers one that is relatively prime with the other
4
4
4
can always be selected.b) Prove that of
10
10
10
consecutive positive integers one that is relatively prime with the other
9
9
9
can always be selected.
123
1
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MMO 123 Moscow MO 1947 remainder x+x^3+x^9 +x^27+x^81+x^243 by x-1
Find the remainder after division of the polynomial
x
+
x
3
+
x
9
+
x
27
+
x
81
+
x
243
x+x^3 +x^9 +x^{27} +x^{81} +x^{243}
x
+
x
3
+
x
9
+
x
27
+
x
81
+
x
243
by
x
−
1
x-1
x
−
1
.