# Problems of the Week

Contribute a problem

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?

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 $$\sum (a+b+p)$$.

×