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Welcome to the Second International Googological Olympiad! Rules are as follows:

• There are 5 questions and a bonus question, worth a total of 50 points.
• Contest ends at 23:59 PST on April 12, 2015.
• Solutions are to be mailed to my e-mail address. It is an account at hotmail.com, and the user name is my name (Royce Peng).
• If anyone needs clarification, please bring it up in the comments. However, please do not spoil any question by giving away the answer or saying something that gives insight into solving the problem. If you cannot do so, or are not sure whether your comment will be a spoiler for the problem, please e-mail me with your question instead.
• Please do not discuss the problems with each other before the due date.
• I will try to respond reasonably promptly to e-mailed solutions, so do not send a solution until you feel that you have done your best with it.
• After I have responded, you may resubmit a solution with a penalty. The penalty is -1 if you misread the problem, and more otherwise. (-2 for a 5 point problem and -3 for a 10 point problem)
• Remember, solutions require you to prove what is asked. Answers alone do not constitute a solution, although they may be worth partial credit.
• You may use any theorem that you find, unless it turns out that the question is merely a special case of the theorem. You may e-mail me if you are in doubt about the use of a theorem or result.
• I reserve the right to amend the rules as needed.

And so without further ado, here is the contest:

Problem 1. (5 points)

Find the number of zeroes in the decimal expansion of $$10 [3] n$$, where $$a [3] b$$ means "$$a$$ in $$b$$ triangles" in Steinhaus-Moser notation. You must express your answer in terms of nonnegative integers with the operations of addition, multiplication, exponentiation, Steinhaus-Moser notation (in the form "$$a [b] c$$"), and the sum and product formulas. For example, the expression

$\sum_{i=1}^n (3^{i+2} + i [3] i)$

would be an acceptable expression.

Problem 2. (5 points)

Find, with proof, the first 10 digits of the googolplexth prime number. (Warning: An asymptotic formula will not be sufficient to solve this problem.)

Problem 3. (10 points)

Define the hyperfactorial by $$H(n) = \prod_{i=1}^n i^i = 1^1 2^2 \ldots n^n$$. Prove that for $$n \ge 1$$,

$\frac{n^{\frac{n^2+n}{2}}}{e^{\frac{n^2-1}{4}}} \le H(n) \le \frac{n^{\frac{n^2+n}{2}}}{e^{\frac{n^2-n}{4}}}$.

Comparable or tighter bounds will receive credit. Weaker bounds will get partial credit.

Problem 4. (10 points)

Call a set $$S$$ of positive integers k-special if the following holds for all $$n \in S, n > k$$:

1. $$n$$ does not divide any other element of $$S$$.

2. $$n$$ divides the product of all smaller elements of $$S$$.

For example, $$\lbrace 1,2,3,4,8 \rbrace$$ is 4-special, as the above rules are satisfied for $$n=8$$, but not 3-special, as $$n=4$$ violates both rules (4 divides 8 and 4 does not divide 1*2*3).

Let $$s(k)$$ be the largest number that is an element of some k-special set. Prove that $$s(5) > 2^{2^{2^{2^{2^{47}}}}}$$. (You do not need to prove existence of $$s(k)$$.)

Problem 5. (10 points)

Define $$S(k)$$ to be sequences from $$\lbrace 1,2,\ldots,k\rbrace$$. (For example, "15243245" is an element of $$S(5)$$. Define $$F(k,n)$$ to be the longest sequence $$\lbrace f_i \rbrace$$ of elements of $$S(k)$$ such that for all $$i$$, $$f_i$$ has length at most $$n+i$$, and for all $$i < j$$, $$f_i$$ is not a subsequence of $$f_j$$. (For example, if $$f_5 = 11$$, then for no $$i > 5$$ could we have $$f_i = 121$$, as 11 is a subsequence of 121.)

Example: the sequence $$22, 121, 1112, 112, 12, 2111111, 211111, 21111, 2111, 211, 21, 2, 11111111111111, \ldots, 1$$ is a sequence for $$F(2,1)$$, so $$F(2,1) \ge 26$$.

Prove that for $$n \ge 1$$, $$F(2,n) > 2 \uparrow^n 4$$.

Bonus Problem. (10 points)

Find nontrivial bounds for $$F(k,n)$$ is Problem 5. In particular, prove that $$F(k,n) > f_{\omega^{k-1}}(n)$$ and $$F(k,1) > f_{\omega^{k-1}}(2)$$, but any interesting bounds will receive credit.