MathDB

Problem 4

Part of 2009 VJIMC

Problems(2)

sequences cover Z

Source: VJIMC 2009 1.4

6/12/2021
Let (an)n=1∞(a_n)_{n=1}^\infty be a sequence of real numbers. We say that the sequence (an)n=1∞(a_n)_{n=1}^\infty covers the set of positive integers if for any positive integer mm there exists a positive integer kk such that ∑n=1∞ank=m\sum_{n=1}^\infty a_n^k=m.
a) Does there exist a sequence of real positive numbers which covers the set of positive integers? b) Does there exist a sequence of real numbers which covers the set of positive integers?
Sequences
partition coloring, prove inequality

Source: VJIMC 2009 2.4

6/12/2021
Let k,m,nk,m,n be positive integers such that 1≤m≤n1\le m\le n and denote S={1,2,…,n}S=\{1,2,\ldots,n\}. Suppose that A1,A2,…,AkA_1,A_2,\ldots,A_k are mm-element subsets of SS with the following property: for every i=1,2,…,ki=1,2,\ldots,k there exists a partition S=S1,i∪S2,i∪…∪Sm,iS=S_{1,i}\cup S_{2,i}\cup\ldots\cup S_{m,i} (into pairwise disjoint subsets) such that
(i) AiA_i has precisely one element in common with each member of the above partition. (ii) Every Aj,j≠iA_j,j\ne i is disjoint from at least one member of the above partition.
Show that k≤(n−1m−1)k\le\binom{n-1}{m-1}.
combinatoricsinequalities