# Problems of the Week

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In this diagram of four concentric circles, the combined area of the yellow regions is the same as the combined area of the orange and white regions.

$$z_1, z_2, z_3,$$ and $$z_4$$ are positive integers such that $$\gcd (z_1,z_2,z_3,z_4) = 1$$.

Must it be true that $$z_1 = z_2 = z_3 = z_4?$$

Diagram not necessarily drawn to scale.

If $$x$$ is chosen uniformly at random on the interval $$(0,1],$$ then what is the probability that $\left\lfloor \frac{\lfloor x\rfloor+1}{x} \right\rfloor \leq 3?$

Notation: $$\lfloor \cdot \rfloor$$ denotes the floor function.

Write down all the integers 1 to 1,000,000 in a row.

• Cross out all of the odd numbers.
• Of the remaining numbers, cross out all of those that are now in even positions (i.e. cross out 4, 8, 12, 16, ...).
• Of the remaining numbers, cross out all of those that are now in odd positions.
• Continue crossing out numbers in this manner, alternating at each stage between even and odd positions, until one number remains.

Let $$A$$ be this last number standing.

Now, write down the numbers 1 to one million again. Repeat the process, but this time with the words "even" and "odd" switched. Let $$B$$ be the last number remaining in this case.

What is the sum $$A+B?$$

A cuboid block of dimensions $$a \times a \times c$$ $$($$with $$a\ll c)$$ and uniform density $$\rho<0.5$$ floats on water. Depending on the density $$\rho$$, there are three different equilibrium possibilities, illustrated in the figure.

A. The bottom side is horizontal.

B. The bottom side makes an angle of $$\alpha<45^{\circ}$$ to the horizontal. Note that there are two possibilities, one with a leftward tilt and the other with a rightward tilt, and these two configurations are equally stable.

C. The block floats so that two bottom sides make $$45^{\circ}$$ to the horizontal.

For densities close to $$\rho=0.5,$$ C correctly describes the stable equilibrium. Configurations B and A follow as the density decreases and approaches $$\rho=0.0$$.

What is the minimum density for configuration C to be stable?

Notes: The density $$\rho$$ is measured relative to the water. For $$\rho>0.5,$$ the stable configurations follow in reverse order, with configuration A being stable for $$\rho$$ approaching 1, except for the fact that the block is much more submerged.

A stationary lawn sprinkler waters a circular region of grass by launching water droplets at an angle of $$\theta=30^\circ$$ above the horizontal in all directions.

The water droplets leave the sprinkler with a range of speeds $$0 \lt v \lt \infty$$ according to the velocity distribution $f_v(v)=\frac{v}{\ v_0^2\ }e^{-v/v_0}.$ The probability that a droplet is launched with a speed between $$v$$ and $$v+dv$$ is $$f_v(v) dv,$$ so the most likely speed is $$v_0$$, the maximum of $$f_v.$$

If the droplets follow parabolic trajectories with acceleration $$g$$ toward the ground, find the probability distribution describing the distance $$r$$ where a droplet lands on the grass.

What is the most likely distance (in meters) from the sprinkler that a droplet lands?

Assumptions:

• Air drag is negligible.
• The height from which the droplets are launched is negligible.
• The most likely speed of a droplet is $$v_0=\SI[per-mode=symbol]{1}{\meter\per\second},$$ and $$g=\SI[per-mode=symbol]{10}{\meter\per\second\squared}.$$
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