MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Kyiv City MO
2024 Kyiv City MO
2024 Kyiv City MO Round 1
2024 Kyiv City MO Round 1
Part of
2024 Kyiv City MO
Subcontests
(5)
Problem 5
4
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Problem 4
3
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Make S nonnegative
For real numbers
a
1
,
a
2
,
…
,
a
200
a_1, a_2, \ldots, a_{200}
a
1
,
a
2
,
…
,
a
200
, we consider the value
S
=
a
1
a
2
+
a
2
a
3
+
…
+
a
199
a
200
+
a
200
a
1
S = a_1a_2 + a_2a_3 + \ldots + a_{199}a_{200} + a_{200}a_1
S
=
a
1
a
2
+
a
2
a
3
+
…
+
a
199
a
200
+
a
200
a
1
. In one operation, you can change the sign of any number (that is, change
a
i
a_i
a
i
to
−
a
i
-a_i
−
a
i
), and then calculate the value of
S
S
S
for the new numbers again. What is the smallest number of operations needed to always be able to make
S
S
S
nonnegative?Proposed by Oleksii Masalitin
Maestro saves algebra
Positive real numbers
a
1
,
a
2
,
…
,
a
2024
a_1, a_2, \ldots, a_{2024}
a
1
,
a
2
,
…
,
a
2024
are arranged in a circle. It turned out that for any
i
=
1
,
2
,
…
,
2024
i = 1, 2, \ldots, 2024
i
=
1
,
2
,
…
,
2024
, the following condition holds:
a
i
a
i
+
1
<
a
i
+
2
a_ia_{i+1} < a_{i+2}
a
i
a
i
+
1
<
a
i
+
2
. (Here we assume that
a
2025
=
a
1
a_{2025} = a_1
a
2025
=
a
1
and
a
2026
=
a
2
a_{2026} = a_2
a
2026
=
a
2
). What largest number of positive integers could there be among these numbers
a
1
,
a
2
,
…
,
a
2024
a_1, a_2, \ldots, a_{2024}
a
1
,
a
2
,
…
,
a
2024
?Proposed by Mykhailo Shtandenko
Fantastic polynomial problem
For a positive integer
n
n
n
, does there exist a permutation of all its positive integer divisors
(
d
1
,
d
2
,
…
,
d
k
)
(d_1 , d_2 , \ldots, d_k)
(
d
1
,
d
2
,
…
,
d
k
)
such that the equation
d
k
x
k
−
1
+
…
+
d
2
x
+
d
1
=
0
d_kx^{k-1} + \ldots + d_2x + d_1 = 0
d
k
x
k
−
1
+
…
+
d
2
x
+
d
1
=
0
has a rational root, if:a)
n
=
2024
n = 2024
n
=
2024
; b)
n
=
2025
n = 2025
n
=
2025
?Proposed by Mykyta Kharin
Problem 3
5
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Problem 2
4
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Problem 1
5
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