= Solution
Assume $n\geq1$ and list the elements of $X$ as $a_1,\ldots,a_n$. Define the <partial sums> $s_j=a_1+\cdots+a_j$ for $1\leq j\leq n$. If one is divisible by $n$, its corresponding nonempty prefix is a suitable $Y$.
Otherwise the $n$ <partial sums> have remainders in the $n-1$ nonzero <residue classes>. The <pigeonhole principle> gives $i<j$ with $s_i\equiv s_j\pmod n$. Thus $s_j-s_i=a_{i+1}+\cdots+a_j$ is divisible by $n$, giving a nonempty <subset> $Y=\{a_{i+1},\ldots,a_j\}$.
Therefore \b[a nonempty $Y$ exists]. This strengthens the literal assertion, which would otherwise allow the trivial <empty set> with sum zero.
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