# Problems of the Week

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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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