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2018-11-12 Advanced


20172017 is a prime number, so the sum of all its positive divisors is 1+2017=2018.1+2017= 2018.

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

n=1+41+51+61+ n=\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+\cdots }}}}

What is the value of n?n?

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

Join the point 00 to each of the other points, and let xkx_k denote the length of the segment joining the points 00 and k.k.

What is the value of 100k=128xk?\displaystyle\left\lfloor 100\prod_{k=1}^{28} x_k \right\rfloor?

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

Given that π=4k=1(1)k+12k1,\displaystyle \pi = 4\sum_{k=1}^{\infty} \frac{\left(-1\right)^{k+1}}{2k-1}, what is the value of

n=1(π4k=1n(1)k+12k1)?\sum_{n=1}^{\infty} \left( \pi - 4\sum_{k=1}^{n} \frac{\left(-1\right)^{k+1}}{2k-1} \right)?


x±x±x±x±\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 xx that guarantees the above expression has a real value?

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

Note: Consider the worst-case scenario, among all possible places where the 20182018 minuses could be.


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