MathDB
Problems
Contests
National and Regional Contests
Russia Contests
239 Open Math Olympiad
2022 239 Open Mathematical Olympiad
2022 239 Open Mathematical Olympiad
Part of
239 Open Math Olympiad
Subcontests
(8)
1
2
Hide problems
Problem 1
A piece is placed in the lower left-corner cell of the
15
×
15
15 \times 15
15
×
15
board. It can move to the cells that are adjacent to the sides or the corners of its current cell. It must also alternate between horizontal and diagonal moves
(
(
(
the first move must be diagonal
)
.
).
)
.
What is the maximum number of moves it can make without stepping on the same cell twice
?
?
?
Polynomial with many different integer roots
Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial
a
99
x
99
+
⋯
+
a
1
x
+
a
0
a_{99}x^{99} + \cdots + a_1x + a_0
a
99
x
99
+
⋯
+
a
1
x
+
a
0
with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?
2
2
Hide problems
Problem 2
Five edges of a tetrahedron are tangent to a sphere. Prove that there are another five edges from this tetrahedron that are also tangent to a
(
(
(
not necessarily the same
)
)
)
sphere.
Geometry with circumscribed quadrilateral
Point
I
I{}
I
is the center of the circle inscribed in the quadrilateral
A
B
C
D
ABCD
A
BC
D
. Prove that there is a point
K
K{}
K
on the ray
C
I
CI
C
I
such that
∠
K
B
I
=
∠
K
D
I
=
∠
B
A
I
\angle KBI=\angle KDI=\angle BAI
∠
K
B
I
=
∠
KD
I
=
∠
B
A
I
.
3
1
Hide problems
Problem 3
Let
A
A
A
be a countable set, some of its countable subsets are selected such that; the intersection of any two selected subsets has at most one element. Find the smallest
k
k
k
for which one can ensure that we can color elements of
A
A
A
with
k
k
k
colors such that each selected subsets exactly contain one element of one of the colors and an infinite number of elements of each of the other colors.
4
2
Hide problems
Problem 4
Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair
(
x
+
y
,
x
)
(x+y,x)
(
x
+
y
,
x
)
or a pair
(
2
x
+
y
+
1
,
x
+
y
+
1
)
(2x+y+1,x+y+1)
(
2
x
+
y
+
1
,
x
+
y
+
1
)
from a pair
(
x
,
y
)
.
(x,y).
(
x
,
y
)
.
At the beginning, the pair
(
1
,
1
)
(1,1)
(
1
,
1
)
is presented on the calculator. Prove that for any natural
n
n
n
there is exactly one pair
(
n
,
k
)
(n,k)
(
n
,
k
)
that can be obtained using a calculator.
Nice graph theory
The degrees of all vertices of a graph are not less than 100 and not more than 200. Prove that its vertices can be divided into connected pairs and triples.
5
1
Hide problems
Problem 5
Prove that there are infinitely many positive integers
k
k
k
such that
k
(
k
+
1
)
(
k
+
2
)
(
k
+
3
)
k(k+1)(k+2)(k+3)
k
(
k
+
1
)
(
k
+
2
)
(
k
+
3
)
has no prime divisor of the form
8
t
+
5.
8t+5.
8
t
+
5.
6
2
Hide problems
Problem 6
239
239
239
wise men stand in a circle near an opaque baobab. The king put on the head of each of these wise men a hat og one of
16
16
16
colors. Each wise men does nor know the color of his hat and can only see the two nearest wise men on each side around the circle. Without communicating, these wise men must at the same time make a guess about the color of their hat
(
(
(
i.e, tell one color
)
)
)
. These wise men were allowed to consult in advance, while they are afraid of being too lucky. What is the maximum
k
k
k
for which, in any arrangement of hats, they can certainly ensure that no more than
k
k
k
wise men guess the color of their hats
?
?
?
Geometry with rectangle
On the side
B
C
BC
BC
of the rectangle
A
B
C
D
ABCD
A
BC
D
, a point
P
P{}
P
is marked so that
∠
A
P
D
=
9
0
∘
\angle APD = 90^\circ
∠
A
P
D
=
9
0
∘
. On the straight line
A
D
AD
A
D
, points
Q
Q{}
Q
and
R
R{}
R
are selected outside the segment
A
D
AD
A
D
such that
A
Q
=
B
P
AQ = BP
A
Q
=
BP
and
C
P
=
D
R
CP = DR
CP
=
D
R
. The circle
ω
\omega
ω
passes through the points
Q
,
D
Q, D
Q
,
D
and the circumcenter of the triangle
P
D
Q
PDQ
P
D
Q
. The circle
γ
\gamma
γ
passes through the points
A
,
R
A, R
A
,
R
and the circumcenter of the triangle
A
P
R
APR
A
PR
. Prove that the radius of one of the circles touching the line
A
D
AD
A
D
and the circles
ω
\omega
ω
and
γ
\gamma
γ
is
2
A
B
2AB
2
A
B
.
7
1
Hide problems
Problem 7
Points
A
,
B
,
C
A,B,C
A
,
B
,
C
are chosen inside the triangle
A
1
B
1
C
1
,
A_{1}B_{1}C_{1},
A
1
B
1
C
1
,
so that the quadrilaterals
B
1
C
B
C
1
,
C
1
A
C
A
1
B_{1}CBC_{1}, C_{1}ACA_{1}
B
1
CB
C
1
,
C
1
A
C
A
1
and
A
1
B
A
B
1
A_{1}BAB_{1}
A
1
B
A
B
1
are inscribed in the circles
Ω
A
,
Ω
B
\Omega _{A}, \Omega _{B}
Ω
A
,
Ω
B
and
Ω
C
,
\Omega _{C},
Ω
C
,
respectively. The circle
Y
A
Y_{A}
Y
A
internally touches the circles
Ω
B
,
Ω
C
\Omega _{B}, \Omega _{C}
Ω
B
,
Ω
C
and externally touches the circle
Ω
A
.
\Omega _{A}.
Ω
A
.
The common interior tangent to the circles
Y
A
Y_{A}
Y
A
and
Ω
A
\Omega _{A}
Ω
A
intersects the line
B
C
BC
BC
at point
A
′
.
A'.
A
′
.
Points
B
′
B'
B
′
and
C
′
C'
C
′
are analogously defined. Prove that points
A
′
,
B
′
A',B'
A
′
,
B
′
and
C
′
C'
C
′
are lying on the same line.
8
2
Hide problems
Problem 8
Prove that there is positive integers
N
N
N
such that the equation
a
r
c
t
a
n
(
N
)
=
∑
i
=
1
2020
a
i
a
r
c
t
a
n
(
i
)
,
arctan(N)=\sum_{i=1}^{2020} a_i arctan(i),
a
rc
t
an
(
N
)
=
i
=
1
∑
2020
a
i
a
rc
t
an
(
i
)
,
does not hold for any integers
a
i
.
a_{i}.
a
i
.
Process of adding rational numbers
There are several rational numbers written on a board. If the numbers
x
x{}
x
and
y
y{}
y
are present on the board, we can add the number
(
x
+
y
)
/
(
1
−
x
y
)
(x+y)/(1-xy)
(
x
+
y
)
/
(
1
−
x
y
)
to it. Prove that there is a rational number that cannot ever appear on the board.