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Find the last three digits of the sum of all positive integers \(x\) such that \[\sqrt{x+\sqrt{1000}} = \sqrt{a} + \sqrt{b}, \]

where \(a\) and \(b\) are positive integers.

The system starts at ...

150 concentric circles with radii 1, 2, 3, . . . , 150 are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by ...

If \((x-5)^2 + (y-4)^2 = 3^2\), maximize \((x-4)^2 + (y-5)^2\).

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