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Union of not almost identical boxes must have large volume

Source: Alibaba Global Math Competition 2021, Problem 10

July 4, 2021
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Problem Statement

In R3\mathbb{R}^3, for a rectangular box Δ\Delta, let 10Δ10\Delta be the box with the same center as Δ\Delta but dilated by 1010. For example, if Δ\Delta is an 1×1×101 \times 1 \times 10 box (hence with Lebesgue measure 1010), then 10Δ10\Delta is the 10×10×10010 \times 10 \times 100 box with the same center and orientation as Δ\Delta.
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If two rectangular boxes Δ1\Delta_1 and Δ2\Delta_2 satisfy Δ1⊂10Δ2\Delta_1 \subset 10\Delta_2 and Δ2⊂10Δ1\Delta_2 \subset 10 \Delta_1, we say that they are almost identical.
Find the largest real number aa such that the following holds for some C=C(a)>0C=C(a)>0:
For every positive integer NN and every collection SS of 1×1×N1 \times 1 \times N boxes in R3\mathbb{R}^3, assuming that (i) ∣S∣=N\vert S\vert=N, (ii) every pair of boxes (Δ1,Δ2)(\Delta_1,\Delta_2) taken from SS are not almost identical, and (iii) the long edge of each box in SS forms an angle π4\frac{\pi}{4} against the xyxy-plane. Then the volume ∣⋃Δ∈SΔ∣≥CNa.\left\vert \bigcup_{\Delta \in S} \Delta\right\vert \ge CN^a.
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