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Spherplex (Posted on 2019-11-25) Difficulty: 3 of 5
A sphere is reconstituted into n identical smaller spheres, keeping the total volume constant.

Let a be the ratio of the total surface area of the smaller spheres to the surface area of the original sphere.

Let b be the ratio of the radius of the original sphere to the radius of each of the smaller spheres.

Let c be the ratio of the volume of the original sphere to the volume of each of the smaller spheres.

What is the minimum integer value of a+b+c?

No Solution Yet Submitted by Danish Ahmed Khan    
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soln | Comment 1 of 3
For the large radius R and small radius r

c = n = R^3/r^3

So, b=n^(1/3) and a=n^(2/3)

and the question becomes:

For what integer n is n^(2/3) + n^(1/3) + n also integer?

Dismissing the trivial case n=1, this next happens at n=8,
where a+b+c is 14

Edited on November 25, 2019, 9:32 am
  Posted by Steven Lord on 2019-11-25 09:26:06

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