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2018-10-08 Advanced

         

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.

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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