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2018-11-12 Advanced

         

\(2017\) is a prime number, so the sum of all its positive divisors is \(1+2017= 2018.\)

Is there another positive integer such that the sum of all its positive divisors is also \(2018?\)

\[ n=\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+\cdots }}}} \]

What is the value of \(n?\)

The diagram shows 29 equidistant points (labeled 0,1,2,...,28) placed on a unit circle centered at point A.

Join the point \(0\) to each of the other points, and let \(x_k\) denote the length of the segment joining the points \(0\) and \(k.\)

What is the value of \(\displaystyle\left\lfloor 100\prod_{k=1}^{28} x_k \right\rfloor?\)

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

Given that \(\displaystyle \pi = 4\sum_{k=1}^{\infty} \frac{\left(-1\right)^{k+1}}{2k-1},\) what is the value of

\[\sum_{n=1}^{\infty} \left( \pi - 4\sum_{k=1}^{n} \frac{\left(-1\right)^{k+1}}{2k-1} \right)?\]


Inspiration

\[\sqrt { x\pm \sqrt { x\pm \sqrt { x\pm \sqrt { x\pm \cdots } } } } \]

Within this infinitely nested radical, exactly 2018 of the \(\pm\) symbols are assigned to be minus \((-)\) signs, with the rest of the \(\pm\) symbols assigned to be plus \((+)\) signs (the minus signs don't need to be consecutive).

What's the minimum positive value of \(x\) that guarantees the above expression has a real value?

Enter your answer in the form \(\left\lfloor 1000x \right\rfloor.\)

Note: Consider the worst-case scenario, among all possible places where the \(2018\) minuses could be.

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