67th Romanian Mathematical Olympiad

A non-empty subset of $Y$ has the sum of its elements $S=113!+s$, where $2\le s\le 2+3+\cdots +15$, whence $2\le s\le 119$.

If $2\le s\le 113$, then $S=113!+s=Ms>s$.

If $s\in \{114,116,118\}$, then $S=M2$.

If $s=115$, then $S=M5$.

If $s=117$, then $S=M3$.

If $s=119$, then $S=M7$.