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2018-10-15 Advanced


Two cities are situated at an angular distance of 4545^\circ with respect to Earth's center.

You are asked to design a tunnel through Earth’s crust connecting the two cities such that it minimizes the time TT to commute between the cities when the train moves only under the influence of gravity.

What is this minimum T?T?

Details and Assumptions:

  • Assume Earth to be a sphere of uniform mass density.
  • RR is the radius of Earth.
  • gg is acceleration due to gravity.
  • Ignore friction.

Select a number of points along the circumference of a circle and connect each pair with a chord. These chords will divide the interior of the circle into a number of regions.

The maximum numbers of regions are shown at right for 1 to 4 points.

What is the maximum number of regions for six points?

Imagine you arrange the thirteen integers 11 to 1313 in a row and label them a1,a2,,a13a_1, a_2, \ldots, a_{13} from left to right under the following conditions:

  • an+2>ana_{n+2} > a_n for n=1,2,,11.n = 1, 2, \ldots, 11.
  • ana_n is either n1,n,n - 1, n, or n+1n + 1 for all n=1,2,,13.n=1, 2, \ldots, 13.

How many such arrangements are possible?


A solid cylinder of radius RR rests on another identical cylinder that rests on the floor in an unstable equilibrium.

If the system is slightly disrupted and there's no sliding between any surfaces, with the cylinders maintaining contact, the angle θ\theta that the line joining the centers make with the vertical satisfies θ˙2=ag(1cosθ)R(b+ccosθdcos2θ),\dot{\theta}^2=\frac{ag(1-\cos\theta)}{R\big(b+c\cos\theta-d\cos^2\theta\big)}, where gg is the acceleration due to gravity and a,b,c,da, b, c, d are positive integers such that the fraction is irreducible.

What is a+b+c+d?a+b+c+d?

A 3×3×33 \times 3 \times 3 magic cube is a three-dimensional array of the consecutive integers 1 1 through 27,27, with the special property that the sum along any row, any column, any pillar, or any of the four space diagonals is equal to the same number.

How many different 3×3×3 3 \times 3 \times 3 magic cubes are there?

Note: Rotations of a certain solution are considered the same solution and therefore not counted.


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