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2017-05-29 Advanced

         

A sphere of uniform mass density ρ\rho and radius a+ba+b has a spherical hole of radius bb drilled a distance aa from its center. The gravitational field is measured at the far side of the hole (at the blue point) and is found to be g0.g_0.

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

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

Find the value of S+M.S+M.

Consider two points: A=(0,0)A=(0,0) which is the center of a unit circle, and B=(0,1)B=(0,1) which lies on that unit circle. Now, you choose a third point CC 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,CA, 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 6×66\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?36?


Note: Below is a 3×33\times 3 grid adhering to the rules above.

p3a+1+pa+1=bp\large p^{3a+1}+p^a+1=b^p

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

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