AoPS Academy
written on the first page and 
empty pages. Suppose in each of the next seconds, Alice can flip to the next page, and write down the sum or product of two numbers (possibly the same) which are already written in her grimoire.
be the largest possible number such that for any 
, Alice can write down the number 
on the last page of her grimoire. Prove that there exists a positive integer 
such that for all 
, we have that ![\[n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.\]](http://latex.artofproblemsolving.com/5/7/e/57e2b7e3d97564665d8de28a3afd74bdcf74ba5b.png)
+2 w
of positive integers for which there exist positive integers 
and such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
such that every collection of consecutive positive integers contains an integer whose sum of digits is divisible by 
. for which he can do this, or prove that none exist.
on one side and a 
on the other. Harry has of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly 
coins showing , then he turns over the th coin from the left; otherwise, all coins show and he stops. For example, if 
the process starting with the configuration 
would be 
, which stops after three operations.
, let 
be the number of operations before Harry stops. For example, 
and 
. Determine the average value of over all 
possible initial configurations .
is fixed and points 
are distributed in such a way that the angle 
(in radians). Cut the circle at points 
How many arcs with different lengths are obtained. ?
