Question 33
Let $S$ be a set of 10 positive integers. Prove that one can find two disjoint subsets $A = \{a_{1}, \ldots , a_{k}\}$ and $B = \{b_{1}, \ldots , b_{k}\}$ of $S$ with $|A| = |B|$ such that the sums
$$x = \frac{1}{a_{1}} +\dots +\frac{1}{a_{k}}$$
and
$$y = \frac{1}{b_{1}} +\dots +\frac{1}{b_{k}}$$
differ by less than 0.01; i.e., $|x - y| \lt 1 / 100$
$$x = \frac{1}{a_{1}} +\dots +\frac{1}{a_{k}}$$
and
$$y = \frac{1}{b_{1}} +\dots +\frac{1}{b_{k}}$$
differ by less than 0.01; i.e., $|x - y| \lt 1 / 100$