Question 28
Let the sum of the elements of a set $X$ be denoted by $S(X)$. How many ways can we divide the numbers $2^1, 2^2, \dots, 2^{10}$ into sets $A$ and $B$ such that the equation
$$
x^2 - S(A)x + S(B) = 0
$$
has a positive integer solution? (The sets $A$ or $B$ may be empty.)
*To divide the numbers $c_1, c_2, \dots, c_n$ into sets $A$ and $B$ means that $A$ and $B$ are disjoint sets and each of the numbers $c_1, c_2, \dots, c_n$ must belong to exactly one of them.*
$$
x^2 - S(A)x + S(B) = 0
$$
has a positive integer solution? (The sets $A$ or $B$ may be empty.)
*To divide the numbers $c_1, c_2, \dots, c_n$ into sets $A$ and $B$ means that $A$ and $B$ are disjoint sets and each of the numbers $c_1, c_2, \dots, c_n$ must belong to exactly one of them.*