HMMT February

First we prove that each subset must consist of elements that have the same residue mod 20. Let a subset consist of elements $a_{1}, \ldots, a_{19}$, and consider two lists of partial sums $$\begin{aligned} & a_{1}, a_{1}+a_{2}, a_{1}+a_{2}+a_{3}, \ldots, a_{1}+a_{2}+\cdots+a_{19} \\ & a_{2}, a_{1}+a_{2}, a_{1}+a_{2}+a_{3}, \ldots, a_{1}+a_{2}+\cdots+a_{19} \end{aligned}$$ The residues mod 20 of the partial sums in each list must be pairwise distinct, otherwise subtracting the sum with less terms from the sum with more terms yields a subset whose sum of elements is 0 $(\bmod 20)$. Since the residues must also be nonzero, each list forms a complete nonzero residue class $\bmod 20$. Since the latter 18 sums in the two lists are identical, $a_{1} \equiv a_{2}(\bmod 20)$. By symmetric arguments, $a_{i} \equiv a_{j}(\bmod 20)$ for any $i, j$. Furthermore this residue $1 \leq r \leq 20$ must be relatively prime to 20, because if $d=\operatorname{gcd}(r, 20)>1$ then any $20 / d$ elements of the subset will sum to a multiple of 20. Hence there are $\varphi(20)=8$ possible residues. Since there are 50 elements in each residue class, the answer is $\binom{50}{19}$. We can see that any such subset whose elements are a relatively prime residue $r(\bmod 20)$ works because the sum of any $1 \leq k \leq 19$ elements will be $k r \neq 0(\bmod 20)$