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Part of 2014 Putnam

Problems(2)

Putnam 2014 A6

Source:

12/7/2014
Let nn be a positive integer. What is the largest kk for which there exist n×nn\times n matrices M1,…,MkM_1,\dots,M_k and N1,…,NkN_1,\dots,N_k with real entries such that for all ii and j,j, the matrix product MiNjM_iN_j has a zero entry somewhere on its diagonal if and only if i≠j?i\ne j?
PutnamalgebrapolynomialvectorfunctionPutnam matricesPutnam 2014
Putnam 2014 B6

Source:

12/8/2014
Let f:[0,1]→Rf:[0,1]\to\mathbb{R} be a function for which there exists a constant K>0K>0 such that ∣f(x)−f(y)∣≤K∣x−y∣|f(x)-f(y)|\le K|x-y| for all x,y∈[0,1].x,y\in [0,1]. Suppose also that for each rational number r∈[0,1],r\in [0,1], there exist integers aa and bb such that f(r)=a+br.f(r)=a+br. Prove that there exist finitely many intervals I1,…,InI_1,\dots,I_n such that ff is a linear function on each IiI_i and [0,1]=⋃i=1nIi.[0,1]=\bigcup_{i=1}^nI_i.
Putnamfunctionslopeintegrationcollege contestsreal analysisPutnam 2014