2018-02-05 Advanced Practice Problems Online

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:

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

Let $A \subset \mathbb Z$ 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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2018-02-05 Advanced