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National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2023 Flanders Math Olympiad
4
4
Part of
2023 Flanders Math Olympiad
Problems
(1)
12 mathematicians separated into 2 clans
Source: 2023 Belgium, VWO Flanders MO p4
3/25/2024
There are
12
12
12
mathematicians living in a village, each of whom belongs to the
2
\sqrt2
2
-clan or belong to the
π
\pi
π
-clan. Moreover every mathematician's birthday is in a different month and every mathematician has an odd number of friends among them the mathematicians. We agree that if mathematician
A
A
A
is a friend of mathematician
B
B
B
, then so is
B
B
B
is a friend of
A
A
A
. On his birthday, every mathematician looks at which clan the majority of his friends belong to, and decides to join that clan until his next birthday. Prove that the mathematicians no longer change clans after a certain point.
combinatorics