MathDB

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Problems(2)

Functional equation with powers of 2

Source:

11/21/2010
Find all functions f:N→Nf:\mathbb{N}\rightarrow \mathbb{N} such that for all m,n∈Nm,n\in\mathbb{N}, (2m+1)f(n)f(2mn)=2mf(n)2+f(2mn)2+(2m−1)2n.(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n.
functionquadraticsalgorithminductionfloor functionalgebra proposedalgebra
Sum of B-entries equals the sum of corresponding A-entries

Source:

11/21/2010
Given real numbers xi,yi(i=1,2,…,n)x_i,y_i (i=1,2,\ldots ,n), let AA be the n×nn\times n matrix given by aij=1a_{ij}=1 if xi≥yjx_i\ge y_j and aij=0a_{ij}=0 otherwise. Suppose BB is a n×nn\times n matrix whose entries are 00 and 11 such that the sum of entries in any row or column of BB equals the sum of entries in the corresponding row or column of AA. Prove that B=AB=A.
linear algebramatrixcombinatorics proposedcombinatorics