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2018-03-26 Intermediate

         

Johnny and Klaus engineer an array of magnets in the following pattern:

Is the magnetic field stronger on side A or B?

A diagonal divides a large, outer square into two equal parts. A smaller square is inscribed in each part. Let the area of the blue square be \(A,\) and let the area of the orange square be \(B.\)

What is \(\frac AB?\)

Which is larger?

\[\begin{align} A &= \dfrac1{2} + \dfrac1{2\times2} + \dfrac1{2\times2\times2} + \dfrac1{2\times2\times2\times2} + \cdots \\ \phantom0\\ B &= \dfrac0{2} + \dfrac1{2\times2} + \dfrac2{2\times2\times2} + \dfrac3{2\times2\times2\times2} + \cdots \end{align}\]

A nomadic tribe in the northern hemisphere moves along the following route every year:

  • In spring, the tribe moves 100 km to the east.
  • In summer, the tribe moves 100 km to the north.
  • In autumn, the tribe moves 98 km to the west.
  • In winter, the tribe moves 100 km to the south.

The tribe reaches its exact starting point from the spring and sets up its winter quarters there.

At what latitude \(\phi\) are the winter quarters (in degrees)? Round to the nearest integer.

Note: Earth's radius is \(R = 6371 \,\text{km}.\)

Lynn has a calculator with only two buttons that perform \(\boxed{+1}\) and \(\boxed{\div 2}\).

She also has a screen with \(9\) significant figures, and it displays \(0\) when she gets the calculator.

If she wants to display \(\pi\) up to the eighth decimal \((3.14159265)\), what is the fewest number of taps she needs to do?

Hint: The first 64 bits of \(\pi\) in binary are given below:

\(\pi \approx\)11.00100100001111110110101010001000100001011010001100001000110100\(_2\)

Note: You may want to use the code environment below.


        

        

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