Question 15
(YUG 1) Let $p$ be a prime and $A=\{a_1,\ldots,a_{p-1}\}$ an arbitrary subset of the natural numbers such that none of its elements is divisible by $p$. Define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to $P=\{0,1,\ldots,p-1\}$ as follows:
(i) if $B=\{a_{i_1},\ldots,a_{i_k}\}\subset A$ and $\sum_{j=1}^k a_{i_j}\equiv n\pmod p$, then $f(B)=n$;
(ii) $f(\varnothing)=0$, $\varnothing$ being the empty set.
Prove that for each $n\in P$ there is a subset $B\subset A$ with $f(B)=n$.
(i) if $B=\{a_{i_1},\ldots,a_{i_k}\}\subset A$ and $\sum_{j=1}^k a_{i_j}\equiv n\pmod p$, then $f(B)=n$;
(ii) $f(\varnothing)=0$, $\varnothing$ being the empty set.
Prove that for each $n\in P$ there is a subset $B\subset A$ with $f(B)=n$.