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Question Number 19547 by dioph last updated on 12/Aug/17

A matrix has N rows and 2k−1   columns. Each column is filled with  M ones and N−M zeros.  A given row j is “cool” if and only if  Σ_(i=1) ^(2k−1) a_(ji)  ≥ k. Find the minimum and  the maximum number of cool rows  for given N, k and M.

$$\mathrm{A}\:\mathrm{matrix}\:\mathrm{has}\:{N}\:\mathrm{rows}\:\mathrm{and}\:\mathrm{2}{k}−\mathrm{1}\: \\ $$$$\mathrm{columns}.\:\mathrm{Each}\:\mathrm{column}\:\mathrm{is}\:\mathrm{filled}\:\mathrm{with} \\ $$$${M}\:\mathrm{ones}\:\mathrm{and}\:{N}−{M}\:\mathrm{zeros}. \\ $$$$\mathrm{A}\:\mathrm{given}\:\mathrm{row}\:{j}\:\mathrm{is}\:``{cool}''\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{\mathrm{2}{k}−\mathrm{1}} {\sum}}{a}_{{ji}} \:\geqslant\:{k}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{maximum}\:\mathrm{number}\:\mathrm{of}\:\mathrm{cool}\:\mathrm{rows} \\ $$$$\mathrm{for}\:\mathrm{given}\:{N},\:{k}\:\mathrm{and}\:{M}. \\ $$

Commented by dioph last updated on 12/Aug/17

For instance, if N=5, k=2, M=2   [(1,0,0),(1,0,0),(0,1,0),(0,1,1),(0,0,1) ]has 1 cool row (min)   [(1,1,0),(1,0,1),(0,1,1),(0,0,0),(0,0,0) ]has 3 cool rows (max)

$$\mathrm{For}\:\mathrm{instance},\:\mathrm{if}\:{N}=\mathrm{5},\:{k}=\mathrm{2},\:{M}=\mathrm{2} \\ $$$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{1}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{1}}\end{bmatrix}\mathrm{has}\:\mathrm{1}\:\mathrm{cool}\:\mathrm{row}\:\left(\mathrm{min}\right) \\ $$$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{1}}&{\mathrm{0}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\end{bmatrix}\mathrm{has}\:\mathrm{3}\:\mathrm{cool}\:\mathrm{rows}\:\left(\mathrm{max}\right) \\ $$

Commented by dioph last updated on 31/Aug/17

anyone?

$$\mathrm{anyone}? \\ $$

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