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 \(N?\)
by
Pi Han Goh
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.
by
Timothy Samson
\[\LARGE n¡=n^{{(n-1)}^{{(n-2)}^{.^{.^{.^{2^{\small1}}}}}}}\]
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.
by
Nick Turtle
\[\sum_{n=1}^\infty \binom{4n}{2n}^{-1}=X+\frac{\pi}{\sqrt{Y}}-\frac{2}{Z\sqrt{Z}}\ln\left(\frac{1+\sqrt{Z}}{2}\right)\]
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:
\[\begin{align} \sum_{n=1}^\infty \binom{4n}{2n}^{-1} &= \frac{1}{\binom{4}{2}} + \frac{1}{\binom{8}{4}} + \frac{1}{\binom{12}{6}} + \cdots \\ &= \frac{1}{6}+\frac{1}{70}+\frac{1}{924}+\cdots. \end{align}\]
by
Digvijay Singh
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?
by
Arjen Vreugdenhil