MathDB

In a

100\times25

rectangle table, fill in a positive real number in each blank. Let the number in the

i

th line, the

j

th column be

x_{i,j}(i=1,2,\cdots,100,j=1,2,\cdots,25)

(shown in Fig.1 ). Then, we rearrange the numbers in each column:

x'_{1,j}\geq x'_{2,j}\geq\cdots\geq x'_{100,j}(j=1,2,\cdots,25)

(shown in Fig.2 ). Find the minumum value of

k

, satisfying: As long as

\sum_{j=1}^{25}x_{i,j}\leq1

for numbers in Fig.1 (

i=1,2,\cdots,100

), then

\sum_{j=1}^{25}x'_{i,j}\leq1

for

i\geq k

in Fig.2. Fig.1\\ \begin{tabular}{|c|c|c|c|} \hline $x_{1,1}$&$x_{1,2}$&$\cdots$&$x_{1,25}$\\ \hline $x_{2,1}$&$x_{2,2}$&$\cdots$&$x_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x_{100,1}$&$x_{100,2}$&$\cdots$&$x_{100,25}$\\ \hline \end{tabular} \qquadFig.2\\ \begin{tabular}{|c|c|c|c|} \hline $x'_{1,1}$&$x'_{1,2}$&$\cdots$&$x'_{1,25}$\\ \hline $x'_{2,1}$&$x'_{2,2}$&$\cdots$&$x'_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x'_{100,1}$&$x'_{100,2}$&$\cdots$&$x'_{100,25}$\\ \hline \end{tabular}

Problem Statement