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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)?
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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 $S+M.$
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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.
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We need to fill each cell in a $6\times6$ grid with a distinct integer from 1 to 36. There are two rules:
Over all valid configurations, how many of the 36 cells could contain the number $36?$
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Note: Below is a $3\times 3$ grid adhering to the rules above.
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$\large p^{3a+1}+p^a+1=b^p$
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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