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National and Regional Contests
North Macedonia Contests
Macedonian Team Selection Test
2022 Macedonian Team Selection Test
Problem 5
Problem 5
Part of
2022 Macedonian Team Selection Test
Problems
(1)
different parities of number of prime divisors of adjacent members of an AP
Source: Macedonian TST 2022, P5
5/21/2022
Given is an arithmetic progression {
a
n
a_n
a
n
} of positive integers. Prove that there exist infinitely many
k
k
k
, such that
ω
(
a
k
)
\omega (a_k)
ω
(
a
k
)
is even and
ω
(
a
k
+
1
)
\omega (a_{k+1})
ω
(
a
k
+
1
)
is odd (
ω
(
n
)
\omega (n)
ω
(
n
)
is the number of distinct prime factors of
n
n
n
).
Proposed
by
Viktor
Simjanoski
and
Nikola
Velov
\textit {Proposed by Viktor Simjanoski and Nikola Velov}
Proposed by Viktor Simjanoski and Nikola Velov
number theory