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2018-04-30 Advanced


S=133+233+333+433++8933S=1^{33} + 2^{33} + 3^{33} + 4^{33} + \cdots + 89^{33}

What is the units digit of S?S?

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

Find A+B+C.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 cm)2 \text{ cm}) that is circular at the top ((with radius 1 cm)1\text{ cm}) but square at the bottom ((with side length 2 cm).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 cm3,\text{cm}^{3}, which can be written as AB+Cπ\frac{A}{B}+C\pi with A,B,CA,B,C integers, AA and BB coprime, and BB positive.

Give the value of A+B+C.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 xx from Earth, as Earth rotates around the Sun. A spacecraft can remain at this point without using any thrust.

What is the distance xx of this point from the Earth in gigameters? (1 gigameter=106 km)\big(1\text{ gigameter}=10^6\text{ km}\big)

Details and Assumptions:

  • The masses of the Earth and the Sun have a ratio of 1:333,000.1: 333,000.
  • The distance between the Earth and the Sun is a150106 km.a \approx 150 \cdot 10^6 \text{ km}.
  • The radii of the Earth and the Sun are negligibly small.

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