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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2017 Mexico National Olympiad
6
6
Part of
2017 Mexico National Olympiad
Problems
(1)
Vote manipulation
Source: Mexico National Olympiad 2017, Problem 6
11/7/2017
Let
n
≥
2
n \geq 2
n
≥
2
and
m
m
m
be positive integers.
m
m
m
ballot boxes are placed in a line. Two players
A
A
A
and
B
B
B
play by turns, beginning with
A
A
A
, in the following manner. Each turn,
A
A
A
chooses two boxes and places a ballot in each of them. Afterwards,
B
B
B
chooses one of the boxes, and removes every ballot from it.
A
A
A
wins if after some turn of
B
B
B
, there exists a box containing
n
n
n
ballots. For each
n
n
n
, find the minimum value of
m
m
m
such that
A
A
A
can guarantee a win independently of how
B
B
B
plays.
combinatorics