Easy One

I used the diophantine solver from the sympy Python-language library known as sympy:

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from sympy.solvers.diophantine import diophantine
from sympy import *

a,b,c=symbols('a b c')
diophantine( a+b+c-1)

It gave me the parametric result

set([(t, -c - t + 1, c)])

't' is clearly the same as 'a'. I can therefore simply evaluate these three expressions for the acceptable values for 'a' and 'c' to obtain all possible acceptable solutions. They are:

-1 1 1

0 1 0

0 0 1

1 1 -1

1 0 0

1 -1 1