MathDB

3

Part of 1994 Polish MO Finals

Problems(2)

number of maps and limit

Source: Problem 3, Polish NO 1994

10/7/2005
kk is a fixed positive integer. Let ana_n be the number of maps ff from the subsets of {1,2,...,n}\{1, 2, ... , n\} to {1,2,...,k}\{1, 2, ... , k\} such that for all subsets A,BA, B of {1,2,...,n}\{1, 2, ... , n\} we have f(A∩B)=min⁡(f(A),f(B))f(A \cap B) = \min (f(A), f(B)). Find lim⁡n→∞ann\lim_{n \to \infty} \sqrt[n]{a_n}.
limitLaTeXfunctionalgebra unsolvedalgebra
n distinct reals and a,b,c,d

Source: Problem 6, Polish NO 1994

10/7/2005
The distinct reals x1,x2,...,xnx_1, x_2, ... , x_n ,(n>3n > 3) satisfy ∑i=1nxi=0\sum_{i=1}^n x_i = 0, ∑i=1nxi2=1\sum_{i=1}^n x_i ^2 = 1. Show that four of the numbers a,b,c,da, b, c, d must satisfy: a+b+c+nabc≤∑i=1nxi3≤a+b+d+nabd a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd .
inequalities unsolvedinequalities