I posted this question many years ago on math stackexchange but it did not get an answer. It had circulated as a puzzle in graduate school.

A disk $D$ of radius $1$ contains disks $D_i$ ($i \ge 1$) of radius $r_i<1$ with pairwise disjoint interiors. Assuming the $D_i$ "use up" the area of $D$ in the sense that $\sum r_i^2=1,$ show the sum of the unsquared radii $\sum r_i$ diverges.