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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.

z1,z2,z3,z_1, z_2, z_3, and z4z_4 are positive integers such that gcd(z1,z2,z3,z4)=1\gcd (z_1,z_2,z_3,z_4) = 1.

Must it be true that z1=z2=z3=z4?z_1 = z_2 = z_3 = z_4?

Diagram not necessarily drawn to scale. Diagram not necessarily drawn to scale.

If xx is chosen uniformly at random on the interval (0,1],(0,1], then what is the probability that x+1x3?\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 AA 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 BB be the last number remaining in this case.

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

A cuboid block of dimensions a×a×ca \times a \times c ((with ac)a\ll c) and uniform density ρ<0.5\rho<0.5 floats on water. Depending on the density ρ\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 α<45\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 4545^{\circ} to the horizontal.

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

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

Notes: The density ρ\rho is measured relative to the water. For ρ>0.5,\rho>0.5, the stable configurations follow in reverse order, with configuration A being stable for ρ\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 θ=30\theta=30^\circ above the horizontal in all directions.

The water droplets leave the sprinkler with a range of speeds 0<v< 0 \lt v \lt \infty according to the velocity distribution fv(v)=v v02 ev/v0.f_v(v)=\frac{v}{\ v_0^2\ }e^{-v/v_0}. The probability that a droplet is launched with a speed between vv and v+dvv+dv is fv(v)dv,f_v(v) dv, so the most likely speed is v0v_0, the maximum of fv.f_v.

If the droplets follow parabolic trajectories with acceleration gg toward the ground, find the probability distribution describing the distance rr 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 v0=1 m/s,v_0=\SI[per-mode=symbol]{1}{\meter\per\second}, and g=10 m/s2.g=\SI[per-mode=symbol]{10}{\meter\per\second\squared}.
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