## Cyclic Lattice Points

For any positive integer \(N \geq 2\), let \(D(N)\) denote the diameter of the smallest circle which passes through exactly \(N\) different lattice points. For example:

- \(D(2) = 1.0\)
- \(D(3) = \frac{5}{3}\sqrt{2} = 2.3570226039551585\ldots\)
- \(D(4) = \sqrt{2} = 1.4142135623730951\ldots\)

Part A | What are the first \(6\) digits of \(D(8)\) after the decimal point? | |
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Part B | What are the first \(6\) digits of \(D(5)\) after the decimal point? |

## Rearrange Letters

If \(W\) is a word consisting of some English letters then let \(X\) be the number of ways to rearrange the letters of \(W\) such that no two adjacent letters are the same. For example if \(W =\) `ROSS' then \(X = 6\) because the valid rearrangements are:

`RSOS', `OSRS', `SROS', `SORS', `SRSO' and `SOSR'

Part A | If \(W = \) `GUSTYGAMES' then find \(X\). | |
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Part B | If \(W = \) `ABRACADABRA' then find \(X\). |