Level
pending

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?

The answer is 979.

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Imagine the pyramid made by the 4 centres and the centre of big sphere.

It has height $\sqrt{500^2 - (100\sqrt{2})^2}=\sqrt{250000-20000}=\sqrt{230000}$

Required height is radius

1 + height of pyramid + radius2 .And thus required answer is $100 + 400 +\lfloor \sqrt{230000} \rfloor = 479+500 = \boxed{979}$