User blog:Wythagoras/Googological problems I.

Each question carries 10 marks, but the questions are roughly ordered from easy to hard.

Post the solutions in the comments under a black with.

1. Find the exact value of \(\frac{(10^{100}-1)!2\cdot H(10^{100})}{(10^{100}!)^{10^{100}}}\)

2. We define the following variant on the block subsequence theorem.

Suppose we have a string x1, x2, ... made of an alphabet of k letters, such that no block of letters xi, ..., xdi is a substring of any later block xj, ..., xdjfor some given d. Call n(k,d) the length of the longest sequence.

Prove n(k,d) > d3, assuming its existence.

3. Find the highest power of 9 that divides \(10^{9^{1000000}}-1\)

4. Find which number is the largest: \(\text{expostfacto}(9)\downarrow^{4}3\) or \(\text{expostfacto}(9\downarrow^{4}3)\).