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area of disjoint circles in a square

Source: miklos schweitzer 1997 q5

September 27, 2021
geometryreal analysis

Problem Statement

Let a1>a2>a3>⋯a_1>a_2>a_3>\cdots be a sequence of real numbers which converges to 0. We put circles of radius a1a_1 into a unit square until no more can fit. (A previously laid circle must not be moved.) Then we put circles of radius a2a_2 in the remaining space until no more can fit, continuing the process for a3a_3,... What can the area covered by the circles be?
a similar problem involving circles in a square: https://artofproblemsolving.com/community/c7h1979044
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