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Problems
Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1941 Moscow Mathematical Olympiad
1941 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(20)
090
1
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MMO 090 Moscow MO 1941 right triangle construction
Construct a right triangle, given two medians drawn to its legs.
089
1
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MMO 089 Moscow MO 1941 locus of midpoints in space
Given two skew perpendicular lines in space, find the set of the midpoints of all segments of given length with the endpoints on these lines.
088
1
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MMO 088 Moscow MO 1941 diophantine x + y = x^2 - xy + y^2
Solve in integers the equation
x
+
y
=
x
2
−
x
y
+
y
2
x + y = x^2 - xy + y^2
x
+
y
=
x
2
−
x
y
+
y
2
.
087
1
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MMO 087 Moscow MO 1941 points and discs of radius 1
On a plane, several points are chosen so that a disc of radius
1
1
1
can cover every
3
3
3
of them. Prove that a disc of radius
1
1
1
can cover all the points.
086
1
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MMO 086 Moscow MO 1941 triangle construction
Given three points
H
1
,
H
2
,
H
3
H_1, H_2, H_3
H
1
,
H
2
,
H
3
on a plane. The points are the reflections of the intersection point of the heights of the triangle
△
A
B
C
\vartriangle ABC
△
A
BC
through its sides. Construct
△
A
B
C
\vartriangle ABC
△
A
BC
.
085
1
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MMO 085 Moscow MO 1941 p^2 mod 12 is 1, where p prime
Prove that the remainder after division of the square of any prime
p
>
3
p > 3
p
>
3
by
12
12
12
is equal to
1
1
1
.
084
1
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MMO 084 Moscow MO 1941 factoring in 2 integer polynomials
a) Find an integer
a
a
a
for which
(
x
−
a
)
(
x
−
10
)
+
1
(x - a)(x - 10) + 1
(
x
−
a
)
(
x
−
10
)
+
1
factors in the product
(
x
+
b
)
(
x
+
c
)
(x + b)(x + c)
(
x
+
b
)
(
x
+
c
)
with integers
b
b
b
and
c
c
c
.b) Find nonzero and nonequal integers
a
,
b
,
c
a, b, c
a
,
b
,
c
so that
x
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
+
1
x(x - a)(x - b)(x - c) + 1
x
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
+
1
factors into the product of two polynomials with integer coefficients.
083
1
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MMO 083 Moscow MO 1941 self-intersecting closed broken line
Consider
△
A
B
C
\vartriangle ABC
△
A
BC
and a point
M
M
M
inside it. We move
M
M
M
parallel to
B
C
BC
BC
until
M
M
M
meets
C
A
CA
C
A
, then parallel to
A
B
AB
A
B
until it meets
B
C
BC
BC
, then parallel to
C
A
CA
C
A
, and so on. Prove that
M
M
M
traverses a self-intersecting closed broken line and find the number of its straight segments.
082
1
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MMO 082 Moscow MO 1941 min no flip parts of a triangle
* Given
△
A
B
C
\vartriangle ABC
△
A
BC
, divide it into the minimal number of parts so that after being flipped over these parts can constitute the same
△
A
B
C
\vartriangle ABC
△
A
BC
.
081
1
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MMO 081 Moscow MO 1941 divide rectangle into different 5,6 squares
a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes.b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.
080
1
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MMO 080 Moscow MO 1941 no of roots sin x = x/10
How many roots does equation
sin
x
=
x
100
\sin x = \frac{x}{100}
sin
x
=
100
x
have?
079
1
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MMO 079 Moscow MO 1941 |x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2
Solve the equation:
∣
x
+
1
∣
−
∣
x
∣
+
3
∣
x
−
1
∣
−
2
∣
x
−
2
∣
=
x
+
2
|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2
∣
x
+
1∣
−
∣
x
∣
+
3∣
x
−
1∣
−
2∣
x
−
2∣
=
x
+
2
.
078
1
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MMO 078 Moscow MO 1941 triangle construction
Given points
M
M
M
and
N
N
N
, the bases of heights
A
M
AM
A
M
and
B
N
BN
BN
of
△
A
B
C
\vartriangle ABC
△
A
BC
and the line to which the side
A
B
AB
A
B
belongs. Construct
△
A
B
C
\vartriangle ABC
△
A
BC
.
077
1
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MMO 077 Moscow MO 1941 integer P(x), with odd values at 0,1, no integer roots
A polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients takes odd values at
x
=
0
x = 0
x
=
0
and
x
=
1
x = 1
x
=
1
. Prove that
P
(
x
)
P(x)
P
(
x
)
has no integer roots.
076
1
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MMO 076 Moscow MO 1941 squares on sides of parallelogram
On the sides of a parallelogram, squares are constructed outwards. Prove that the centers of these squares are vertices of a square.
074
1
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MMO 074 Moscow MO 1941 locus of midpoints of chords
A point
P
P
P
lies outside a circle. Consider all possible lines drawn through
P
P
P
so that they intersect the circle. Find the locus of the midpoints of the chords — segments the circle intercepts on these lines.
073
1
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MMO 073 Moscow MO 1941 congruent triangles wanted
Given a quadrilateral, the midpoints
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
of its consecutive sides, and the midpoints of its diagonals,
P
P
P
and
Q
Q
Q
. Prove that
△
B
C
P
=
△
A
D
Q
\vartriangle BCP = \vartriangle ADQ
△
BCP
=
△
A
D
Q
.
075
1
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MMO 075 Moscow MO 1941 product of 4 consecutive +1 a perfect square
Prove that
1
1
1
plus the product of any four consecutive integers is a perfect square.
072
1
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MMO 072 Moscow MO 1941 \overline 523abc divisible by 7, 8,9
Find the number
523
a
b
c
‾
\overline {523abc}
523
ab
c
divisible by
7
,
8
7, 8
7
,
8
and
9
9
9
.
071
1
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MMO 071 Moscow MO 1941 triangle construction
Construct a triangle given its height and median — both from the same vertex — and the radius of the circumscribed circle.