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Canada National Olympiad
2021 Canada National Olympiad
5
5
Part of
2021 Canada National Olympiad
Problems
(1)
Canadian MO 2021 P5
Source:
3/12/2021
Nina and Tadashi play the following game. Initially, a triple
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of nonnegative integers with
a
+
b
+
c
=
2021
a+b+c=2021
a
+
b
+
c
=
2021
is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer
k
k
k
and one of the three entries on the board; then the player increases the chosen entry by
k
k
k
and decreases the other two entries by
k
k
k
. A player loses if, on their turn, some entry on the board becomes negative.Find the number of initial triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
for which Tadashi has a winning strategy.
combinatorics