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Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2020 Latvia Baltic Way TST
6
6
Part of
2020 Latvia Baltic Way TST
Problems
(1)
Blink and you miss the solution
Source: Latvia TST for Baltic Way 2020 P6
10/22/2020
For a natural number
n
≥
3
n \ge 3
n
≥
3
we denote by
M
(
n
)
M(n)
M
(
n
)
the minimum number of unit squares that must be coloured in a
6
×
n
6 \times n
6
×
n
rectangle so that any possible
2
×
3
2 \times 3
2
×
3
rectangle (it can be rotated, but it must be contained inside and cannot be cut) contains at least one coloured unit square. Is it true that for every natural
n
≥
3
n \ge 3
n
≥
3
the number
M
(
n
)
M(n)
M
(
n
)
can be expressed as
M
(
n
)
=
p
n
+
k
n
3
M(n)=p_n+k_n^3
M
(
n
)
=
p
n
+
k
n
3
, where
p
n
p_n
p
n
is a prime and
k
n
k_n
k
n
is a natural number?
combinatorics