= Solution
First justify the finite bound used in the hint. Fix $k$. If no finite $T$ forced a <monochromatic> positive solution of $\sum_i a_i x_i=0$, consider the rooted <tree> whose level $t$ consists of the solution-free <finite colorings> of $[t]$, with restriction as the predecessor map. Every level is nonempty and every vertex has at most $k$ children. The <König infinity lemma> gives an infinite branch, hence a <finite coloring> of all <positive integers> with no such solution, contradicting the <partition regular matrix> hypothesis. This is the <compactness bound for partition regularity>.
Choose such a $T$ and let $S=\operatorname{lcm}(1,\ldots,T)$, the <least common multiple>. For a given <finite coloring> $\chi$ of the <positive integers>, pull it back to $[T]$ by
$$
\widetilde\chi(t)=\chi(S/t).
$$
Each $S/t$ is a <positive integer> in $[S]$. The defining property of $T$ gives $x_1,\ldots,x_n\in[T]$ of one color under $\widetilde\chi$, with $\sum_i a_i x_i=0$. Set $y_i=S/x_i$. Their colors under $\chi$ agree, and
$$
\boxed{\sum_{i=1}^n\frac{a_i}{y_i}
=\frac1S\sum_{i=1}^n a_i x_i=0.}
$$
Thus <reciprocal partition regularity> follows. This construction is an involution on the divisors of $S$; it permits repeated coordinates and never requires a reciprocal of a <positive integer> to itself be integral without the common scaling factor $S$.
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