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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.

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    $\begingroup$ Compare Section 4 in web.math.princeton.edu/sarnak/InternalApollonianPackings09.pdf, where the divergence of $\sum r_i$ is credited to O. Wesler, “An infinite packing theorem for spheres,” PAMS Vol. 11, pp. 324-326, (1960). $\endgroup$
    – JHM
    Commented Feb 6, 2021 at 15:18
  • $\begingroup$ @JHM Thanks for that link. Gives me a place to look. $\endgroup$
    – coffeemath
    Commented Feb 6, 2021 at 15:24
  • $\begingroup$ @JHM Now that I looked at it, Wesler's article looks direct enough that I will likely understand it on further study. I'll accept your answer below. $\endgroup$
    – coffeemath
    Commented Feb 6, 2021 at 15:29
  • $\begingroup$ Very interesting question. $\endgroup$
    – JHM
    Commented Feb 6, 2021 at 15:47

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Proven by O. Wesler, “An infinite packing theorem for spheres,” PAMS Vol. 11, pp. 324-326, (1960).

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