2024 SSMO Relay Round 5 Problems/Problem 1

Problem

Let the super factorial be defined on positive integers as Find the largest positive integer such that that there are exactly positive integers such that has fewer than trailing zeroes.

Solution

Let be the number of trailing zeroes has and denote the number of trailing zeroes has. Clearly, Consider the following table:

\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} n&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\\hline g(n)&0&0&0&0&1&1&1&1&1&2&2&2&2&2\\ f(n)&0&0&0&0&1&2&3&4&5&7&9&11&13&15\\ \end{array}

\begin{array}{ccccccccccccc}

 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & . & 0 \\
 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & . & 0 \\
 &   & 1 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & . & 0 \\
 &   &   & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & . & 0 \\
 &   &   &   & 2 & 5 & 0 & 0 & 0 & 0 & 0 & . & 0 \\
 &   &   &   &   & 3 & 0 & 0 & 0 & 0 & 0 & . & 0 \\
 &   &   &   &   &   & 3 & 5 & 0 & 0 & 0 & . & 0 \\
 &   &   &   &   &   &   & 4 & 0 & 0 & 0 & . & 0 \\
 &   &   &   &   &   &   &   & 4 & 5 & 0 & . & 0 \\

+ & & & & & & & & & 5 & 0 & . & 0 \\

 & 6 & 1 & 7 & 2 & 8 & 3 & 9 & 5 & 0 & 5 & . & 5 \\

\end{array}

It is easy to see that for exactly integers such that has fewer than trailing zeroes. Thus, the answer is .

~SMO_Team