The Cuboid must be a Cube?

Geometry Level 1

True or False?

For positive reals \(a,\) \(b,\) and \(c,\) if \(3abc = a^3 + b^3 + c^3,\) then it is necessarily true that \(a=b=c.\)

Note: One can interpret this geometrically. Consider a cuboid with side lengths \(a,b,c,\) and three cubes with respective side lengths \(a,b,c.\) If the sum of the volumes of the cubes is the same as three times the volume of the cuboid, then is the cuboid necessarily a cube?


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