Subcontests
(3)number of maps and limit
k is a fixed positive integer. Let an be the number of maps f from the subsets of {1,2,...,n} to {1,2,...,k} such that for all subsets A,B of {1,2,...,n} we have f(A∩B)=min(f(A),f(B)). Find limn→∞nan. n distinct reals and a,b,c,d
The distinct reals x1,x2,...,xn ,(n>3) satisfy ∑i=1nxi=0, ∑i=1nxi2=1. Show that four of the numbers a,b,c,d must satisfy:
a+b+c+nabc≤i=1∑nxi3≤a+b+d+nabd. jugs with water
m,n are relatively prime. We have three jugs which contain m, n and m+n liters. Initially the largest jug is full of water. Show that for any k in {1,2,...,m+n} we can get exactly k liters into one of the jugs.