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Serbia Contests
Serbia National Math Olympiad
2020 Serbia National Math Olympiad
6
6
Part of
2020 Serbia National Math Olympiad
Problems
(1)
Onedimensional board game.
Source: 2020 Serbian MO, Problem 6
9/26/2020
We are given a natural number
k
k
k
. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute
n
n
n
coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves:
(
i
)
(i)
(
i
)
We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other.
(
i
i
)
(ii)
(
ii
)
We choose a field with at least
2
2
2
coins on it, and we transfer one coin from the chosen field to the
k
−
t
h
k-\mathrm{th}
k
−
th
field on the left , and one coin from the chosen field to the
k
−
t
h
k-\mathrm{th}
k
−
th
field on the right.
(
a
)
\mathbf{(a)}
(
a
)
If
n
≤
k
+
1
n\leq k+1
n
≤
k
+
1
, prove that we can play only finitely many moves.
(
b
)
\mathbf{(b)}
(
b
)
For which values of
k
k
k
we can choose a natural number
n
n
n
and distribute
n
n
n
coins on the given board such that we can play infinitely many moves.
Game Theory
combinatorics