User blog:Wythagoras/Problems for the First International Googological Olympiad

Rules

 * You can send your solutions to my mailadress.
 * If you don't know my mailadress, you can leave a message on my talk page with your mailadress, so that I can mail you. If you don't want that, make an appointment, and you can tell it in chat.
 * Everyone can participate.
 * You can participate anonymously, but at least give a pseudonym.
 * If you have questions about the problems you can ask in comment.
 * Please do not discuss the problems before the due date.
 * You need to give proofs, only answers aren't enough.
 * There are four problems worth seven points each.
 * The maximal score is seven points. The minimal score is zero points.
 * Due date: February 22, 2015, 23:59 UTC.
 * If you've sent your solutions, you can leave a comment here. I'll look to them as soon as possible.
 * You can use any theorem or result somewhere on the internet, however, make sure that I can find it, otherwise you'll lose points.
 * I determine the final number of points, there is no discussion possible about that.
 * There are no prizes, exepted for honour of course.
 * I can change rules if needed.

Problems
1. Find a number \(n\) such that \(\frac{\sigma(n)}{n}\) is larger than 1,000. (\(\sigma(n)\) is the sum of the divisors of \(n\)).

2. Proof that \(SCG(n)\) is even for every \(n \in \mathbb N\). (empty graph is counted)

3. Determine all quadruples of numbers (a,b,c,d) such that \(a \rightarrow b \rightarrow c \rightarrow d > \{a,b,c,d\}\)

4. A function f is k-exponential iff f(n) = E(O(n))#k, using Hyper-E.

For example, a function f is 3-exponential iff \(f(n) = 2^{2^{2^{O(n)}}}\).

Find, for every \(k \in \mathbb N\), a function f such that \(f^n(n)\) is a k-exponential function.