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LFM Pure
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Problem Text
To celebrate the opening of the financial year the finance minister of Genland flings a Slihing, a circular coin of radius $a$ cm, where $0 < a < 1$, onto a large board divided into squares by two sets of parallel lines 2 cm apart. If the coin does not cross any line, or if the coin covers an intersection, the tax on yaks remains unchanged. Otherwise the tax is doubled. Show that, in order to raise most tax, the value of $a$ should be \[\left(1+{\displaystyle \frac{\pi}{4}}\right)^{-1}.\] If, indeed, $a=\left(1+{\displaystyle \frac{\pi}{4}}\right)^{-1}$ and the tax on yaks is 1 Slihing per yak this year, show that its expected value after $n$ years will have passed is \[ \left(\frac{8+\pi}{4+\pi}\right)^{n}.\]
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