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## Rules

• 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.
• 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.