AoPS Academy
books are given as prizes to students (each students gets one book). In how many ways can the organisers give these prizes if they have copies of one book and 
other books each in one copy?
. Show that there exists 
such that![\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\]](http://latex.artofproblemsolving.com/4/d/2/4d269e5b682be6778e89237aa91da5e0399042a7.png)
so that the sum of its edges' lengths is minumum.
find, in terms of , the minimum value of 
for which there is a set of 
distinct positive integers that has sum greater than but every subset of size has sum at most 
be a set that intersects all non-finite integer arithmetic progressions, 
be the set of prime divisors of 
and 
be the set of prime divisors of 
. Suppose 
. Prove that 
, 
be a set that intersects all non-finite integer arithmetic progressions, be the set of prime divisors of and be the set of prime divisors of . Suppose . Prove that , 
). Maybe the solution by angiland can be adapted but I'm not sure.
