A sphere of uniform mass density \(\rho\) and radius \(a+b\) has a spherical hole of radius \(b\) drilled a distance \(a\) from its center. The gravitational field is measured at the far side of the hole (at the blue point) and is found to be \(g_0.\)

What will the field measure at the near side of the hole (at the green point)?

Let \(x, y, z\) be real numbers satisfying \(x+y+z=15\) and \(xy+yz+zx=72\). Let the minimum value of \(x\) be \(S,\) and the maximum value \(M\).

Find the value of \(S+M.\)

Consider two points: \(A=(0,0)\) which is the center of a unit circle, and \(B=(0,1)\) which lies on that unit circle. Now, you choose a third point \(C\) inside the circle uniformly at random.

What is the probability that you will be able to draw a square such that all three points \(A, B, C\) lie on two adjacent sides of the square?

Please submit your answer to two decimal places.

We need to fill each cell in a \(6\times6\) grid with a distinct integer from 1 to 36. There are two rules:

- Every pair of consecutive numbers are in adjacent cells that share an edge.
- Any two cells containing a multiple of 4
**cannot**share an edge nor a vertex.

Over all valid configurations, how many of the 36 cells could contain the number \(36?\)

\(\)

**Note**: Below is a \(3\times 3\) grid adhering to the rules above.

\[\large p^{3a+1}+p^a+1=b^p\]

Find all solutions to the Diophantine equation above, where \(a\) and \(b\) are positive integers and \(p\) is an odd prime, and enter your answer as \(\sum (a+b+p)\).

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