Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1987 based on Distance and Spheres.

## Distance and Sphere – AIME I, 1987

What is the largest possible distance between two points, one on the sphere of radius 19 with center (-2,-10,5) and the other on the sphere of radius 87 with center (12,8,-16)?

- is 107
- is 137
- is 840
- cannot be determined from the given information

**Key Concepts**

Angles

Algebra

Spheres

## Check the Answer

But try the problem first…

Answer: is 137.

AIME I, 1987, Question 2

Geometry Vol I to Vol IV by Hall and Stevens

## Try with Hints

First hint

The distance between the center of the spheres is \(\sqrt{(12-(-2)^{2}+(8-(-10))^{2}+(-16-5)^{2}}\)

Second Hint

=\(\sqrt{14^{2}+18^{2}+21^{2}}\)=31

Final Step

The largest possible distance=sum of the two radii+distance between the centers=19+87+31=137.

## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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