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Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1940 Moscow Mathematical Olympiad
1940 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(17)
062-
1
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MMO 062- Moscow MO 1940 4digit perfect sqaure wanted
Find a four-digit number that is perfect square and such that the first two digits are the same and the last two as well.
070
1
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MMO 070 Moscow MO 1940 2^x - x^2 is divisible by 7
How many positive integers
x
x
x
less than
10000
10 000
10000
are there such that
2
x
−
x
2
2^x - x^2
2
x
−
x
2
is divisible by
7
7
7
?
069
1
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MMO 069 Moscow MO 1940 \frac{a_1}{a_2}+ ... + \frac{a_n}{a_1} \ge n
Let
a
1
,
.
.
.
,
,
a
n
a_1, ...,, a_n
a
1
,
...
,,
a
n
be positive numbers. Prove the inequality:
a
1
a
2
+
a
2
a
3
+
a
3
a
4
+
.
.
.
+
a
n
−
1
a
n
+
a
n
a
1
≥
n
\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_4}+ ... +\frac{a_{n-1}}{a_n}+ \frac{a_n}{a_1} \ge n
a
2
a
1
+
a
3
a
2
+
a
4
a
3
+
...
+
a
n
a
n
−
1
+
a
1
a
n
≥
n
068
1
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MMO 068 Moscow MO 1940 triangle construction by 3 points
The center of the circle circumscribing
△
A
B
C
\vartriangle ABC
△
A
BC
is mirrored through each side of the triangle and three points are obtained:
O
1
,
O
2
,
O
3
O_1, O_2, O_3
O
1
,
O
2
,
O
3
. Reconstruct
△
A
B
C
\vartriangle ABC
△
A
BC
from
O
1
,
O
2
,
O
3
O_1, O_2, O_3
O
1
,
O
2
,
O
3
if everything else is erased.
067
1
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MMO 067 Moscow MO 1940 compare 300! with 100^{300}
Which is greater:
300
!
300!
300
!
or
10
0
300
100^{300}
10
0
300
?
066
1
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MMO 066 Moscow MO 1940 curve’s self-intersections unfolidng an infinite cone
* Given an infinite cone. The measure of its unfolding’s angle is equal to
α
\alpha
α
. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.
065
1
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MMO 065 Moscow MO 1940 x^2 + y^2 is divisible by 7
How many pairs of integers
x
,
y
x, y
x
,
y
are there between
1
1
1
and
1000
1000
1000
such that
x
2
+
y
2
x^2 + y^2
x
2
+
y
2
is divisible by
7
7
7
?
064
1
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MMO 064 Moscow MO 1940 tile the plane with irregular quadrilaterals
How does one tile a plane, without gaps or overlappings, with the tiles equal to a given irregular quadrilateral?
062
1
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MMO 062 Moscow MO 1940 \overline {abc}=a!+b! +c!
Find all
3
3
3
-digit numbers
a
b
c
‾
\overline {abc}
ab
c
such that
a
b
c
‾
=
a
!
+
b
!
+
c
!
\overline {abc} = a! + b! + c!
ab
c
=
a
!
+
b
!
+
c
!
.
061
1
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MMO 061 Moscow MO 1940 locus with distances from 2 lines
Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.
060
1
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MMO 060 Moscow MO 1940 circle equidistant from 4 points
Construct a circle equidistant from four points on a plane. How many solutions are there?
059
1
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MMO 059 Moscow MO 1940 206788$-th digit of 123456789101112131415...
Consider all positive integers written in a row:
123456789101112131415...
123456789101112131415...
123456789101112131415...
Find the
206788
206788
206788
-th digit from the left.
058
1
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MMO 058 Moscow MO 1940 system (x^3 + y^3)(x^2 + y^2) = 2b^5, x + y = b
Solve the system
{
(
x
3
+
y
3
)
(
x
2
+
y
2
)
=
2
b
5
x
+
y
=
b
\begin{cases} (x^3 + y^3)(x^2 + y^2) = 2b^5 \\ x + y = b \end{cases}
{
(
x
3
+
y
3
)
(
x
2
+
y
2
)
=
2
b
5
x
+
y
=
b
in
C
C
C
057
1
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MMO 057 Moscow MO 1940 no of solutions, circle tangent to line and circle
Draw a circle that has a given radius
R
R
R
and is tangent to a given line and a given circle. How many solutions does this problem have?
056
1
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MMO 056 Moscow MO 1940 zeros of 100!
How many zeros does
100
!
100!
100
!
have at its end in the usual decimal representation?
055
1
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MMO 055 Moscow MO 1940 days a raft floats the river
It takes a steamer
5
5
5
days to go from Gorky to Astrakhan downstream the Volga river and
7
7
7
days upstream from Astrakhan to Gorky. How long will it take for a raft to float downstream from Gorky to Astrakhan?
054
1
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MMO 054 Moscow MO 1940 factor (b - c)^3 + (c - a)^3 + (a - b)^3.
Factor
(
b
−
c
)
3
+
(
c
−
a
)
3
+
(
a
−
b
)
3
(b - c)^3 + (c - a)^3 + (a - b)^3
(
b
−
c
)
3
+
(
c
−
a
)
3
+
(
a
−
b
)
3
.