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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2024 Ukraine National Mathematical Olympiad
2024 Ukraine National Mathematical Olympiad
Part of
Ukraine National Mathematical Olympiad
Subcontests
(8)
Problem 8
3
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Oleksii and Solomiya go pro gaming on a grid
Oleksii and Solomiya play the following game on a square
6
n
×
6
n
6n\times 6n
6
n
×
6
n
, where
n
n
n
is a positive integer. Oleksii in his turn places a piece of type
F
F
F
, consisting of three cells, on the board. Solomia, in turn, after each move of Oleksii, places the numbers
0
,
1
,
2
0, 1, 2
0
,
1
,
2
in the cells of the figure that Oleksii has just placed, using each of the numbers exactly once. If two of Oleksii's pieces intersect at any moment (have a common square), he immediately loses. Once the square is completely filled with numbers, the game stops. In this case, if the sum of the numbers in each row and each column is divisible by
3
3
3
, Solomiya wins, and otherwise Oleksii wins. Who can win this game if the figure of type
F
F
F
is: a) a rectangle ; b) a corner of three cells?Proposed by Oleksii Masalitin
Almost Hamiltonian path
There are
2024
2024
2024
cities in a country, some pairs of which are connected by bidirectional flights. For any distinct cities
A
,
B
,
C
,
X
,
Y
,
Z
A, B, C, X, Y, Z
A
,
B
,
C
,
X
,
Y
,
Z
, it is possible to fly directly from some of the cities
A
,
B
,
C
A, B, C
A
,
B
,
C
to some of the cities
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
. Prove that it is possible to plan a route
T
1
→
T
2
→
…
→
T
2022
T_1\to T_2 \to \ldots \to T_{2022}
T
1
→
T
2
→
…
→
T
2022
that passes through
2022
2022
2022
distinct cities.Proposed by Lior Shayn
Generalization of 10.7
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with integer coefficients, such that for each of them there exists a positive integer
N
N
N
, such that for any positive integer
n
≥
N
n\geq N
n
≥
N
, number
P
(
n
)
P(n)
P
(
n
)
is a positive integer and a divisor of
n
!
n!
n
!
.Proposed by Mykyta Kharin
Problem 7
4
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Problem 6
4
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Problem 5
3
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Best Ukrainian Author strikes with another genius algebra...
Real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are such that
a
2
+
c
−
b
c
=
b
2
+
a
−
c
a
=
c
2
+
b
−
a
b
a^2+c-bc = b^2+a-ca = c^2+b-ab
a
2
+
c
−
b
c
=
b
2
+
a
−
c
a
=
c
2
+
b
−
ab
Does it follow that
a
=
b
=
c
a=b=c
a
=
b
=
c
?Proposed by Mykhailo Shtandenko
Maestro composed an algebra so good we used it in two grades!
For real numbers
a
,
b
,
c
,
d
∈
[
0
,
1
]
a, b, c, d \in [0, 1]
a
,
b
,
c
,
d
∈
[
0
,
1
]
, find the largest possible value of the following expression:
a
2
+
b
2
+
c
2
+
d
2
−
a
b
−
b
c
−
c
d
−
d
a
a^2+b^2+c^2+d^2-ab-bc-cd-da
a
2
+
b
2
+
c
2
+
d
2
−
ab
−
b
c
−
c
d
−
d
a
Proposed by Mykhailo Shtandenko
A bit of casework but imo pleasant
You are given some
12
12
12
non-zero, not necessarily distinct real numbers. Find all positive integers
k
k
k
from
1
1
1
to
12
12
12
, such that among these numbers you can always choose
k
k
k
numbers whose sum has the same sign as their product, that is, either both the sum and the product are positive, or both are negative.Proposed by Anton Trygub
Problem 4
4
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Problem 3
4
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Problem 2
4
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Problem 1
3
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Pairwise Sums
Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from
0
0
0
to
9
9
9
appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote?Proposed by Oleksiy Masalitin
Make equal. Fast!
Solomiya wrote the numbers
1
,
2
,
…
,
2024
1, 2, \ldots, 2024
1
,
2
,
…
,
2024
on the board. In one move, she can erase any two numbers
a
,
b
a, b
a
,
b
from the board and write the sum
a
+
b
a+b
a
+
b
instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this?Proposed by Oleksiy Masalitin
NT equations make a huge comeback
Find all pairs
a
,
b
a, b
a
,
b
of positive integers, for which
(
a
,
b
)
+
3
[
a
,
b
]
=
a
3
−
b
3
(a, b) + 3[a, b] = a^3 - b^3
(
a
,
b
)
+
3
[
a
,
b
]
=
a
3
−
b
3
Here
(
a
,
b
)
(a, b)
(
a
,
b
)
denotes the greatest common divisor of
a
,
b
a, b
a
,
b
, and
[
a
,
b
]
[a, b]
[
a
,
b
]
denotes the least common multiple of
a
,
b
a, b
a
,
b
.Proposed by Oleksiy Masalitin