Let a,b,c and d be four positive integers ,
a < b < c < d
a + b + c + d = 40
You can get any positive integer up to 40, Using a,b,c and d (each integer you can use one time or zero) , and the operations { + , - } only !
For example : if d = 15 ,
● You can get 40 by : a + b + c + d
● Yoy can get 25 by : a + b + c
● You can get 10 by : a + b + c - d
So, what is : (2ad - bc)
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The key is to write the integers up to 40 in base 3:
Then, every integer can be written using sums and differences of powers of 3, so $\boxed{1,3,9,27}$ by the following rules that are applied to each digit going from right to left
(A few examples are written in a comment to this solution)
So, $(a,b,c,d)=(1,3,9,27)$ are integers that can make all integers up to 40 and they satisfy the given conditions because $1<3<9<27$ and $1+3+9+27=40$ .
This means that the answer is $2 \cdot 1 \cdot 27 - 3 \cdot 9 = \boxed{27}$ .