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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Kyiv City MO
2021 Kyiv City MO
2021 Kyiv City MO Round 1
9.4
9.4
Part of
2021 Kyiv City MO Round 1
Problems
(1)
Number of same colored numbers
Source: Kyiv City MO 2021 Round 1, Problem 9.4
12/21/2023
You are given a positive integer
k
k
k
and not necessarily distinct positive integers
a
1
,
a
2
,
a
3
,
…
,
a
k
a_1, a_2 , a_3 , \ldots, a_k
a
1
,
a
2
,
a
3
,
…
,
a
k
. It turned out that for any coloring of all positive integers from
1
1
1
to
2021
2021
2021
in one of the
k
k
k
colors so that there are exactly
a
1
a_1
a
1
numbers of the first color,
a
2
a_2
a
2
numbers of the second color,
…
\ldots
…
, and
a
k
a_k
a
k
numbers of the
k
k
k
-th color, there is always a number
x
∈
{
1
,
2
,
…
,
2021
}
x \in \{1, 2, \ldots, 2021\}
x
∈
{
1
,
2
,
…
,
2021
}
, such that the total number of numbers colored in the same color as
x
x
x
is exactly
x
x
x
. What are the possible values of
k
k
k
?Proposed by Arsenii Nikolaiev
combinatorics