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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.

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