## 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 on my talk page, 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.
- If there are any mistakes in problems, I can fix them and ask you to solve the problem again.
- Please answer the following questions for each problem and post them at the end of your solutions. Give a number form 1 to 10, where 1 is not at all, and 10 is perfectly.
- Was the problem hard?
- Was the problem original?
- Was the problem relevant to googology?

- If you've made a stupid mistake that costs a lot of points, I can send you an email with the mistake back. Both attempts will be listed. The following penalties are given:
- 1 point if you misread the question.
- 2 points if you've made another mistake.

- If there is a draw the person with the least number of attempts wins.
- If there is still a draw there is a draw.

## Problems

**Problem 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 \(n\)).

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

**Problem 3. **Determine all quadruples of numbers (a,b,c,d) such that

\[a \rightarrow b \rightarrow c \rightarrow d > \{a,b,c,d\}\]

**Problem 4. **A function f is k-exponential iff f(n) = E(Θ(n))#k, using Hyper-E. For example, a function f is 3-exponential iff \(f(n) = 2^{2^{2^{\Theta(n)}}}\). Find, for every \(k \in \mathbb N\), a function f such that \(f^n(n)\) is a k-exponential function. Give the function in terms of: addition, multiplication, exponentation, factorial, logarithm, iterated logarithm, tetration, subtraction and division.

## Problem fixes

- Problem 4, fixed thanks to LP9.
- Problem 2, fixed thanks to LP9.