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2013 PUMaC Geometry B3

Source:

November 22, 2013
geometry3D geometry

Problem Statement

Consider all planes through the center of a 2×2×22\times2\times2 cube that create cross sections that are regular polygons. The sum of the cross sections for each of these planes can be written in the form ab+ca\sqrt b+c, where bb is a square-free positive integer. Find a+b+ca+b+c.