Question 33
Prove that the set of integers $\{0, 1, 2, \dots, 2^n - 1\}$ can be partitioned into $n + 1$ disjoint subsets $A_0, A_1, \dots, A_n$ such that both of the following hold:
a) If $k + l = n$, then the subsets $A_k$ and $A_l$ have the same number of elements.
b) If $s$ and $t$ are non-negative integers and $s + t \le n$, then for an arbitrary element $z$ in the set $A_{s+t}$, there exist elements $x$ and $y$ from the sets $A_s$ and $A_t$, respectively, such that $x + y = z$.
a) If $k + l = n$, then the subsets $A_k$ and $A_l$ have the same number of elements.
b) If $s$ and $t$ are non-negative integers and $s + t \le n$, then for an arbitrary element $z$ in the set $A_{s+t}$, there exist elements $x$ and $y$ from the sets $A_s$ and $A_t$, respectively, such that $x + y = z$.