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Problems
Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
2017 Moscow Mathematical Olympiad
2017 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(11)
11
1
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Table with digits
There is one nonzero digit in every cell of
2017
×
2017
2017\times 2017
2017
×
2017
table. On the board we writes
4034
4034
4034
numbers that are rows and columns of table. It is known, that
4033
4033
4033
numbers are divisible by prime
p
p
p
and last is not divisible by
p
p
p
. Find all possible values of
p
p
p
.Example for
2
×
2
2\times2
2
×
2
. If table is |1|4| |3|7|. Then numbers on board are
14
,
37
,
13
,
47
14,37,13,47
14
,
37
,
13
,
47
10
1
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Geometry problem
Point
D
D
D
lies in
△
A
B
C
\triangle ABC
△
A
BC
and
B
D
=
C
D
BD=CD
B
D
=
C
D
,
∠
B
D
C
=
120
\angle BDC=120
∠
B
D
C
=
120
. Point
E
E
E
lies outside
A
B
C
ABC
A
BC
and
A
E
=
C
E
,
∠
A
E
C
=
60
AE=CE,\angle AEC=60
A
E
=
CE
,
∠
A
EC
=
60
. Points
B
B
B
and
E
E
E
lies on different sides of
A
C
AC
A
C
.
F
F
F
is midpoint
B
E
BE
BE
. Prove, that
∠
A
F
D
=
90
\angle AFD=90
∠
A
F
D
=
90
9
1
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Detective interrogates
There are
80
80
80
peoples, one of them is murderer, and other one is witness of crime. Every day detective interrogates some peoples from this group. Witness will says about crime only if murderer will not be in interrogatory with him. It is enough
12
12
12
days to find murderer ?
8
1
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Logarithms
Are there such
x
,
y
x,y
x
,
y
that
lg
(
x
+
y
)
=
lg
x
lg
y
\lg{(x+y)}=\lg x \lg y
l
g
(
x
+
y
)
=
l
g
x
l
g
y
and
lg
(
x
−
y
)
=
lg
x
lg
y
\lg{(x-y)}=\frac{\lg x}{\lg y}
l
g
(
x
−
y
)
=
l
g
y
l
g
x
?
7
1
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Progressions
(
a
i
)
,
(
b
i
)
(a_i),(b_i)
(
a
i
)
,
(
b
i
)
are nonconstant arithmetic and geometric progressions.
a
1
=
b
1
,
a
2
/
b
2
=
2
,
a
4
/
b
4
=
8
a_1=b_1,a_2/b_2=2,a_4/b_4=8
a
1
=
b
1
,
a
2
/
b
2
=
2
,
a
4
/
b
4
=
8
Find
a
3
/
b
3
a_3/b_3
a
3
/
b
3
.
6
1
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Gangsters wars
There are
36
36
36
gangsters bands.And there are war between some bands. Every gangster can belongs to several bands and every 2 gangsters belongs to different set of bands. Gangster can not be in feuding bands. Also for every gangster is true, that every band, where this gangster is not in, is in war with some band, where this gangster is in. What is maximum number of gangsters in city?
5
1
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Unit cube
8
8
8
points lie on the faces of unit cube and form another cube. What can be length of edge of this cube?
4
1
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Cyclists on circle
3 cyclists rides on track in form circle with length
300
300
300
meters in one direction. Every has constant speed,and speeds are different. Photographer want to make photoshoot with 3 cyclists. It is possible if they will be on the part of track with length
d
d
d
meters. Find minimum
d
d
d
such that it is possible.
3
1
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Roots in polynomials
Let
x
0
x_0
x
0
- is positive root of
x
2017
−
x
−
1
=
0
x^{2017}-x-1=0
x
2017
−
x
−
1
=
0
and
y
0
y_0
y
0
- is positive root of
y
4034
−
y
=
3
x
0
y^{4034}-y=3x_0
y
4034
−
y
=
3
x
0
a) Compare
x
0
x_0
x
0
and
y
0
y_0
y
0
b) Find tenth digit after decimal mark in decimal representation of
∣
x
0
−
y
0
∣
|x_0-y_0|
∣
x
0
−
y
0
∣
2
1
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Incircle geometry
ω
\omega
ω
is incircle of
△
A
B
C
\triangle ABC
△
A
BC
touch
A
C
AC
A
C
in
S
S
S
. Point
Q
Q
Q
lies on
ω
\omega
ω
and midpoints of
A
Q
AQ
A
Q
and
Q
C
QC
QC
lies on
ω
\omega
ω
. Prove that
Q
S
QS
QS
bisects
∠
A
Q
C
\angle AQC
∠
A
QC
1
1
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Number permutations
Find minimum number
n
n
n
that: 1)
80
∣
n
80|n
80∣
n
2) we can permute 2 different numbers in
n
n
n
to get
m
m
m
and
80
∣
m
80|m
80∣
m