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

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

Angles

Algebra

Spheres

But try the problem first...

Answer: is 137.

Source

Suggested Reading

AIME I, 1987, Question 2

Geometry Vol I to Vol IV by Hall and Stevens

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.

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

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