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2018-02-05 Advanced

         

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

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

If a periodic non-zero function f(x)f(x) exists such that 3f(x)=f(x1)+f(x+1),\sqrt{3}f(x) = f(x - 1) + f(x + 1), which of these options could be its fundamental period?

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

n¡=n(n1)(n2)...21\LARGE n¡=n^{{(n-1)}^{{(n-2)}^{.^{.^{.^{2^{\small1}}}}}}}

If n¡ is defined as above, what are the last two digits of the number 2018¡?2018¡?

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

n=1(4n2n)1=X+πY2ZZln(1+Z2)\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 XX, YY, and ZZ. Find XYZ\sqrt{XYZ}.

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

n=1(4n2n)1=1(42)+1(84)+1(126)+=16+170+1924+.\begin{aligned} \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{aligned}

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

  • If aa is an element of A,A, then a-a is also an element of AA.

  • If aa is an element of AA, then a+Na+N is also an elements of AA.

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

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


Bonus: Generalize. If NN = 2e23e35e52^{e_2}\cdot 3^{e_3}\cdot 5^{e_5}\cdots is the prime factorization of NN, how many non-trivial remarkable sets of type NN exist? What do they look like?

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