Virginia Mathematics Teacher Fall 2016

So l ut i ons t o Spr i ng 2016 HEXA Cha l l enge Probl ems The inequalities could be rewritten in the following way:

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Let’s multiply all of the inequalities, then :

August Challenge: Four spheres are all tangent to a plane. Three spheres with the radii 2 units, 1 unit, and 1 unit respectively, are standing on the plane around the fourth sphere which is smaller. If all four of the spheres are tangent to each other, what is the radius of the smaller sphere? Note that If two spheres with radii R and r are on the same plane and touch each other, the distance between the points at which the spheres are touching the plane can be expressed through the Pythagorean theorem Then, all the arrangement of the four sphere centers O1, O2, O3 and O4 could be projected on a plane as points A, B, C, and D respectively. The projection of the centers on the plane is the figure below. Where dis­ tance AB is the distance between the projected centers. Below on the left is the side view

Consider the top view, of the four spheres with centers O1, O2, O3, and O4 (above right). We can see that

The solution to this equation can then be calculated as follows:

We have to choose the smaller root because x must be less than 1. Therefore

Virginia Mathematics Teacher vol. 43, no. 1

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