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Problems
Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
2018 Moscow Mathematical Olympiad
2018 Moscow Mathematical Olympiad
Part of
Moscow Mathematical Olympiad
Subcontests
(11)
11
1
Hide problems
Painting ball
Ivan want to paint ball. Ivan can put ball in the glass with some paint, and then one half of ball will be painted. Ivan use
5
5
5
glasses to paint glass competely. Prove, that one glass was not needed, and Ivan can paint ball with
4
4
4
glasses, putting ball in it by same way.
10
1
Hide problems
Some geometry
A
B
C
ABC
A
BC
is acute-angled triangle,
A
A
1
,
C
C
1
AA_1,CC_1
A
A
1
,
C
C
1
are altitudes.
M
M
M
is centroid.
M
M
M
lies on circumcircle of
A
1
B
C
1
A_1BC_1
A
1
B
C
1
. Find all values of
∠
B
\angle B
∠
B
9
1
Hide problems
Big number trigonometry
x
x
x
and
y
y
y
are integer
5
5
5
-digits numbers, such that in the decimal notation, all ten digits are used exactly once. Also
tan
x
−
tan
y
=
1
+
tan
x
tan
y
\tan{x}-\tan{y}=1+\tan{x}\tan{y}
tan
x
−
tan
y
=
1
+
tan
x
tan
y
, where
x
,
y
x,y
x
,
y
are angles in degrees. Find maximum of
x
x
x
8
1
Hide problems
Dominos on field
2018
×
2018
2018\times 2018
2018
×
2018
field is covered with
1
×
2
1 \times 2
1
×
2
dominos, such that every
2
×
2
2 \times 2
2
×
2
or
1
×
4
,
4
×
1
1 \times 4,4 \times 1
1
×
4
,
4
×
1
figure is not covered by only two dominos. Can be covered more than
99
%
99\%
99%
of field ?
7
1
Hide problems
Equation with logarithms
x
3
+
(
log
2
5
+
log
3
2
+
log
5
3
)
x
=
(
log
2
3
+
log
3
5
+
log
5
2
)
x
2
+
1
x^3+(\log_2{5}+\log_3{2}+\log_5{3})x=(\log_2{3}+\log_3{5}+\log_5{2})x^2+1
x
3
+
(
lo
g
2
5
+
lo
g
3
2
+
lo
g
5
3
)
x
=
(
lo
g
2
3
+
lo
g
3
5
+
lo
g
5
2
)
x
2
+
1
6
3
Hide problems
Bad wiring in big house
There is house with
2
n
2^n
2
n
rooms and every room has one light bulb and light switch. But wiring was connected wrong, so light switch can turn on light in some another room. Master want to find what switch connected to every light bulb. He use next practice: he send some workers in the some rooms, then they turn on switches in same time, then they go to master and tell him, in what rooms light bulb was turned on. a) Prove that
2
n
2n
2
n
moves is enough to find, how switches are connected to bulbs. b) Is
2
n
−
1
2n-1
2
n
−
1
moves always enough ?
Divide square
We divide
999
×
999
999\times 999
999
×
999
square into the angles with
3
3
3
cells. Prove, that number of ways is divided by
2
7
2^7
2
7
.( Angle is a figure, that we can get if we remove one cell from
2
×
2
2 \times 2
2
×
2
square).
Club members
There are
2018
2018
2018
peoples. We call the group of people as "club" if all members of same "club" are all friends, but not friends with a nonmember of "club". Prove, that we can divide peoples for
90
90
90
rooms, such that no one room has all members of some "club".
5
2
Hide problems
Geometry with hexagon
On the sides of the convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
into the outer side were built equilateral triangles
A
B
C
1
ABC_1
A
B
C
1
,
B
C
D
1
BCD_1
BC
D
1
,
C
D
E
1
CDE_1
C
D
E
1
,
D
E
F
1
DEF_1
D
E
F
1
,
E
F
A
1
EFA_1
EF
A
1
and
F
A
B
1
FAB_1
F
A
B
1
. The triangle
B
1
D
1
F
1
B_1D_1F_1
B
1
D
1
F
1
is equilateral too. Prove that, the triangle
A
1
C
1
E
1
A_1C_1E_1
A
1
C
1
E
1
is also equilateral.
Coloring triangle
We have a blue triangle. In every move, we divide the blue triangle by angle bisector to
2
2
2
triangles and color one triangle in red. Prove, that after some moves we color more than half of the original triangle in red.
4
3
Hide problems
Sum of cubes
Are there natural solution of
a
3
+
b
3
=
1
1
2018
a^3+b^3=11^{2018}
a
3
+
b
3
=
1
1
2018
?
Fixed point
A
B
C
D
ABCD
A
BC
D
is convex and
A
B
∦
C
D
,
B
C
∦
D
A
AB\not \parallel CD,BC \not \parallel DA
A
B
∥
C
D
,
BC
∥
D
A
.
P
P
P
is variable point on
A
D
AD
A
D
. Circumcircles of
△
A
B
P
\triangle ABP
△
A
BP
and
△
C
D
P
\triangle CDP
△
C
D
P
intersects at
Q
Q
Q
. Prove, that all lines
P
Q
PQ
PQ
goes through fixed point.
Arrangement of ones and zeroes
We call the arrangement of
n
n
n
ones and
m
m
m
zeros around the circle as good, if we can swap neighboring zero and one in such a way that we get an arrangement, that differs from the original by rotation. For what natural
m
m
m
and
n
n
n
does a good arrangement exist?
3
3
Hide problems
Polynomial equation
Are there such natural
n
n
n
, that exist polynomial of degree
n
n
n
and with
n
n
n
different real roots, and a)
P
(
x
)
P
(
x
+
1
)
=
P
(
x
2
)
P(x)P(x+1)=P(x^2)
P
(
x
)
P
(
x
+
1
)
=
P
(
x
2
)
b)
P
(
x
)
P
(
x
+
1
)
=
P
(
x
2
+
1
)
P(x)P(x+1)=P(x^2+1)
P
(
x
)
P
(
x
+
1
)
=
P
(
x
2
+
1
)
Some geometry
O
O
O
is circumcircle and
A
H
AH
A
H
is the altitude of
△
A
B
C
\triangle ABC
△
A
BC
.
P
P
P
is the point on line
O
C
OC
OC
such that
A
P
⊥
O
C
AP \perp OC
A
P
⊥
OC
. Prove, that midpoint of
A
B
AB
A
B
lies on the line
H
P
HP
H
P
.
Number of roots
a
1
,
a
2
,
.
.
.
,
a
k
a_1,a_2,...,a_k
a
1
,
a
2
,
...
,
a
k
are positive integers and
1
a
1
+
1
a
2
+
.
.
.
+
1
a
k
>
1
\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}>1
a
1
1
+
a
2
1
+
...
+
a
k
1
>
1
. Prove that equation
[
n
a
1
]
+
[
n
a
2
]
+
.
.
.
+
[
n
a
k
]
=
n
[\frac{n}{a_1}]+[\frac{n}{a_2}]+...+[\frac{n}{a_k}]=n
[
a
1
n
]
+
[
a
2
n
]
+
...
+
[
a
k
n
]
=
n
has no more than
a
1
∗
a
2
∗
.
.
.
∗
a
k
a_1*a_2*...*a_k
a
1
∗
a
2
∗
...
∗
a
k
postivie integer solutions in
n
n
n
.
2
3
Hide problems
Gluing cube
There is tetrahedron and square pyramid, both with all edges equal
1
1
1
. Show how to cut them into several parts and glue together from these parts a cube (without voids and cracks, all parts must be used)
Coloring square
In there
2018
×
2018
2018\times 2018
2018
×
2018
square cells colored in white or black. It is known, that exists
10
×
10
10 \times 10
10
×
10
square with only white cells and
10
×
10
10\times 10
10
×
10
square with only black cells. For what minimal
d
d
d
always exists square
10
×
10
10\times 10
10
×
10
such that the number of black and white cells differs by no more than
d
d
d
?
Triangle from sticks
We have
4
4
4
sticks. It is known, that for every
3
3
3
sticks we can build a triangle with the same area. Is it true, that sticks have the same length?
1
3
Hide problems
Trinomial and derivative
The graphs of a square trinomial and its derivative divide the coordinate plane into four parts. How many roots does this square trinomial has?
Square of number
Is there a number in the decimal notation of the square which has a sequence of digits "
2018
2018
2018
"?
Easy algebra
a
1
,
a
2
,
.
.
.
,
a
81
a_1,a_2,...,a_{81}
a
1
,
a
2
,
...
,
a
81
are nonzero,
a
i
+
a
i
+
1
>
0
a_i+a_{i+1}>0
a
i
+
a
i
+
1
>
0
for
i
=
1
,
.
.
.
,
80
i=1,...,80
i
=
1
,
...
,
80
and
a
1
+
a
2
+
.
.
.
+
a
81
<
0
a_1+a_2+...+a_{81}<0
a
1
+
a
2
+
...
+
a
81
<
0
. What is sign of
a
1
∗
a
2
∗
.
.
.
∗
a
81
a_1*a_2*...*a_{81}
a
1
∗
a
2
∗
...
∗
a
81
?