AoPS Academy
having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer 
of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation 
. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win?
+1 w
that satisfies:![\[
\sum_{i=1}^{n} a_{\left\lfloor \frac{n}{i} \right\rfloor} = n^k
\]](http://latex.artofproblemsolving.com/b/b/e/bbe7fe81e366fe390be5d17193d1581cc40dbece.png)
be a positive integer. Prove that for all integers 
, we have:![\[
\frac{c^{a_n} - c^{a_{n-1}}}{n} \in \mathbb{Z}
\]](http://latex.artofproblemsolving.com/0/4/4/04464c1a350e3f9a929ab51534018498680068fc.png)
be real positive numbers such that: 
. Prove that: 
, and let the spheres be 
. We can assume that are each tangent to three faces of the sphere, and are in corners 
respectively.
scale factor 
that takes 
and 
, so that 
is inscribed in 
. Translate by 
to 
, which circumscribes 
. Note that 
, so 
from which the result follows.
has all edges of length less than 
,
.
.
- locus of centers of all spheres with diameters 
,
.
.
where 
-
,
so 
,