# Problems of the Week

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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 $$\sqrt{3}f(x) = f(x - 1) + f(x + 1),$$ which of these options could be 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.

$\LARGE n¡=n^{{(n-1)}^{{(n-2)}^{.^{.^{.^{2^{\small1}}}}}}}$

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

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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