Given that a+b=c, a+c=d and b=c+d, where d is a positive integer, what is the greatest value of a?

a + b  = c       ⇒    b  = c - a    (1)

a + c  = d

c + d  = b    (3)

Equating  (1) and (3)

c + d  =  c - a

d  = - a

So

a  =  -d

We can prove this

a + c  = d

-d + c  = d

c = 2d

And

c + d  = b

2d + d = b

3d = b

So....using the original equations with the substitutions...

a + b  =  c  ⇒   -d + 3d =  2d

a + c  = d ⇒  -d + 2d  =  d 

c + d  = b  ⇒  2d + d  = 3d