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Contests
National and Regional Contests
Ukraine Contests
Other Ukrainian Contests
Kyiv Mathematical Festival
2021 Kyiv Mathematical Festival
2021 Kyiv Mathematical Festival
Part of
Kyiv Mathematical Festival
Subcontests
(5)
5
3
Hide problems
Bilbo's triangle
Bilbo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any
n
n
n
digits, and in other rows he always writes
0
0
0
under consecutive equal digits and writes
1
1
1
under consecutive distinct digits. (An example of a triangle for
n
=
5
n=5
n
=
5
is shown below.) In how many ways can Bilbo fill the topmost row for
n
=
100
n=100
n
=
100
so that each of
n
n
n
rows of the triangle contains even number of ones?
1
0
1
1
0
1
1
0
1
0
1
1
1
0
1
\begin{smallmatrix}1\,0\,1\,1\,0\\1\,1\,0\,1\\0\,1\,1\\1\,0\\1\end{smallmatrix}
1
0
1
1
0
1
1
0
1
0
1
1
1
0
1
(O. Rudenko and V. Brayman)
Frodo's triangle
Frodo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any
n
n
n
digits, and in other rows he always writes
0
0
0
under consecutive equal digits and writes
1
1
1
under consecutive distinct digits. (An example of a triangle for
n
=
5
n=5
n
=
5
is shown below.) In how many ways can Frodo fill the topmost row for
n
=
100
n=100
n
=
100
so that each of
n
n
n
rows of the triangle contains odd number of ones?
1
0
1
1
0
1
1
0
1
0
1
1
1
0
1
\begin{smallmatrix}1\,0\,1\,1\,0\\1\,1\,0\,1\\0\,1\,1\\1\,0\\1\end{smallmatrix}
1
0
1
1
0
1
1
0
1
0
1
1
1
0
1
(O. Rudenko and V. Brayman)
Prove that SD=IE
Let
ω
\omega
ω
be the circumcircle of a triangle
A
B
C
ABC
A
BC
(
A
B
≠
A
C
{AB\ne AC}
A
B
=
A
C
),
I
I
I
be the incenter,
P
P
P
be the point on
ω
\omega
ω
for which
∠
A
P
I
=
9
0
∘
,
\angle API=90^\circ,
∠
A
P
I
=
9
0
∘
,
S
S
S
be the intersection point of lines
A
P
AP
A
P
and
B
C
,
BC,
BC
,
W
W
W
be the intersection point of line
A
I
AI
A
I
and
ω
.
\omega.
ω
.
Line which passes through point
W
W
W
orthogonally to
A
W
AW
A
W
meets
A
P
AP
A
P
and
B
C
BC
BC
at points
D
D
D
and
E
E
E
respectively. Prove that
S
D
=
I
E
.
SD=IE.
S
D
=
I
E
.
(Ye. Azarov)
4
1
Hide problems
collection of 63 integers
Find all collections of
63
63
63
integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal. (O. Rudenko)
3
3
Hide problems
prove that points are collinear
Let
ω
\omega
ω
be the circumcircle of a triangle
A
B
C
ABC
A
BC
(
A
B
>
A
C
AB>AC
A
B
>
A
C
),
E
E
E
be the midpoint of the arc
A
C
AC
A
C
which does not contain point
B
,
B,
B
,
аnd
F
F
F
the midpoint of the arc
A
B
AB
A
B
which does not contain point
C
.
C.
C
.
Lines
A
F
AF
A
F
and
B
E
BE
BE
meet at point
P
,
P,
P
,
line
C
F
CF
CF
and
A
E
AE
A
E
meet at point
R
,
R,
R
,
and the tangent to
ω
\omega
ω
at point
A
A
A
meets line
B
C
BC
BC
at point
Q
.
Q.
Q
.
Prove that points
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
are collinear. (M. Kurskiy)
prove that points are concyclic
Let
A
D
AD
A
D
be the altitude,
A
E
AE
A
E
be the median, and
O
O
O
be the circumcenter of a triangle
A
B
C
.
ABC.
A
BC
.
Points
X
X
X
and
Y
Y
Y
are selected inside the triangle such that
∠
B
A
X
=
∠
C
A
Y
,
\angle BAX=\angle CAY,
∠
B
A
X
=
∠
C
A
Y
,
O
X
⊥
A
X
,
OX\perp AX,
OX
⊥
A
X
,
and
O
Y
⊥
A
Y
.
OY\perp AY.
O
Y
⊥
A
Y
.
Prove that points
D
,
E
,
X
,
Y
D,E,X,Y
D
,
E
,
X
,
Y
are concyclic. (M. Kurskiy)
collection of n integers
Is it true that for every
n
≥
2021
n\ge 2021
n
≥
2021
there exist
n
n
n
integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal? (O. Rudenko)
2
3
Hide problems
hedgehogs divide
In 11 cells of a square grid there live hedgehogs. Every hedgehog divides the number of hedgehogs in its row by the number of hedgehogs in its column. Is it possible that all the hedgehogs get distinct numbers? (V.Brayman)
hedgehogs multiply
In several cells of a square grid there live hedgehogs. Every hedgehog multiplies the number of hedgehogs in its row by the number of hedgehogs in its column. Is it possible that all the hedgehogs get distinct numbers?
three variable inequality
Let
a
,
b
,
c
≥
0
a,b,c\ge0
a
,
b
,
c
≥
0
and
a
+
b
+
c
=
3.
a+b+c=3.
a
+
b
+
c
=
3.
Prove that
(
3
a
−
b
c
)
(
3
b
−
a
c
)
(
3
c
−
a
b
)
≤
8.
(3a-bc)(3b-ac)(3c-ab)\le8.
(
3
a
−
b
c
)
(
3
b
−
a
c
)
(
3
c
−
ab
)
≤
8.
(O. Rudenko)
1
2
Hide problems
points and arithmetic progression
Is it possible to mark four points on the plane so that the distances between any point and three other points form an arithmetic progression? (V. Brayman)
solve in integers
Solve equation
(
3
a
−
b
c
)
(
3
b
−
a
c
)
(
3
c
−
a
b
)
=
1000
(3a-bc)(3b-ac)(3c-ab)=1000
(
3
a
−
b
c
)
(
3
b
−
a
c
)
(
3
c
−
ab
)
=
1000
in integers. (V.Brayman)