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 $z_1 = z_2 = z_3 = z_4?$
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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.
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Write down all the integers 1 to 1,000,000 in a row.
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 $A+B?$
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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.
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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:
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