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2018-02-05 Advanced

         

I have \(N+1\) numbers. If I sum any \(N\) of them, the sum is divisible by \(N\).

If one of the numbers is divisible by \(N\), must all of the other numbers also be divisible by \(N?\)

If a periodic non-zero function \(f(x)\) exists such that \(\sqrt{3}f(x) = f(x - 1) + f(x + 1),\) what is its fundamental period?

Note: A function \(g(x)\) is periodic if there exists some \(T\) such that \(g(x) = g(x + T)\) for all \(x.\) The fundamental period is the minimum positive \(T\) such that this property holds.

Define \(n¡\) to be \[\LARGE n¡=n^{{(n-1)}^{{(n-2)}^{.^{.^{.^{2^{\small1}}}}}}}.\] What are the last two digits of the number \(2018¡?\)

Note: \(\large ¡\) is the factorial notation \(!\) turned upside down. Note that \(\large ¡\) keeps exponentiating while \(!\) keeps multiplying.

\[\sum_{n=1}^\infty \binom{4n}{2n}^{-1}=X+\frac{\pi}{\sqrt{Y}}-\frac{2}{Z\sqrt{Z}}\ln\left(\frac{1+\sqrt{Z}}{2}\right)\]

The equation above holds true for rational numbers \(X\), \(Y\), and \(Z\). Find \(\sqrt{XYZ}\).

Note: \(\binom{\cdot}{\cdot}\) is the binomial coefficient. The first few terms of the series are as follows:

\[\begin{align} \sum_{n=1}^\infty \binom{4n}{2n}^{-1} &= \frac{1}{\binom{4}{2}} + \frac{1}{\binom{8}{4}} + \frac{1}{\binom{12}{6}} + \cdots \\ &= \frac{1}{6}+\frac{1}{70}+\frac{1}{924}+\cdots. \end{align}\]

Let \(A \subset \mathbb Z\) be a non-trivial set of integers \((\)i.e. \(A \not= \emptyset\) and \(A \not= \mathbb Z).\) We call such a set \(A\) a "remarkable set of type \(N\)" if it has the following properties:

  • If \(a\) is an element of \(A,\) then \(-a\) is also an element of \(A\).

  • If \(a\) is an element of \(A\), then \(a+N\) is also an elements of \(A\).

  • If \(a,b\) are elements of \(A\) (not necessarily different), then \(a + 2b\) is also an element of \(A\).

How many (non-trivial) remarkable sets of type 18 are there?


Bonus: Generalize. If \(N\) = \(2^{e_2}\cdot 3^{e_3}\cdot 5^{e_5}\cdots\) is the prime factorization of \(N\), how many non-trivial remarkable sets of type \(N\) exist? What do they look like?

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