Four spheres of radii 100 each rest on a horizontal plane H such that their centers form a square of side 200. A sphere of radius 400 rests over them.
Let d be the largest possible distance of a point on the larger sphere from the plane H. What is integral part of d?
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Imagine the pyramid made by the 4 centres and the centre of big sphere.
It has height 5 0 0 2 − ( 1 0 0 2 ) 2 = 2 5 0 0 0 0 − 2 0 0 0 0 = 2 3 0 0 0 0
Required height is radius 1 + height of pyramid + radius 2 .
And thus required answer is 1 0 0 + 4 0 0 + ⌊ 2 3 0 0 0 0 ⌋ = 4 7 9 + 5 0 0 = 9 7 9