2018-04-30 Advanced

         

\[S=1^{33} + 2^{33} + 3^{33} + 4^{33} + \cdots + 89^{33}\]

What is the units digit of \(S?\)

Assume that Earth is a perfect sphere with radius \(R.\) The time \(T\) an object takes to fall to the ground from rest from a height of \(R\) above the ground is given by \[T=\sqrt{\frac Rg}\left(A+\frac {\pi^B}C\right),\] where \(A,B,C\) are positive integers.

Find \(A+B+C.\)

Note: Consider only gravity, and forget about atmospheric effects, air resistance, etc.

After studying various 3D shapes and finding formulas for their volumes, I challenged my students to invent a new shape. Lindsay created a shape \((\)with height \(2 \text{ cm})\) that is circular at the top \((\)with radius \(1\text{ cm})\) but square at the bottom \((\)with side length \(2\text{ cm}).\) Lindsay created this shape from a paraboloid: sliced four times parallel to the paraboloid's axis and two times perpendicular to the paraboloid's axis. Lindsay's shape is pictured below.

Find the volume of Lindsay's shape in \(\text{cm}^{3},\) which can be written as \(\frac{A}{B}+C\pi\) with \(A,B,C\) integers, \(A\) and \(B\) coprime, and \(B\) positive.

Give the value of \(A+B+C.\)

Four nails are randomly fixed on a circular soft board.

Emma takes a red elastic rubber band, stretches it around the nails, and lets go.

What is the probability that the rubber band does not take the shape of a quadrilateral?


Source: Putnam 2006 A-6

There is a special point between Sun and Earth where a spacecraft can be parked, so that it always remains directly between them, at a fixed distance \(x\) from Earth, as Earth rotates around the Sun. A spacecraft can remain at this point without using any thrust.

What is the distance \(x\) of this point from the Earth in gigameters? \(\big(1\text{ gigameter}=10^6\text{ km}\big)\)

Details and Assumptions:

  • The masses of the Earth and the Sun have a ratio of \(1: 333,000.\)
  • The distance between the Earth and the Sun is \(a \approx 150 \cdot 10^6 \text{ km}.\)
  • The radii of the Earth and the Sun are negligibly small.
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