MathDB

In each cell of a matrix

n\times n

a number from a set

\{1,2,\ldots,n^2\}

is written — in the first row numbers

1,2,\ldots,n

, in the second n\plus{}1,n\plus{}2,\ldots,2n and so on. Exactly

n

of them have been chosen, no two from the same row or the same column. Let us denote by

a_i

a number chosen from row number

i

. Show that: \frac{1^2}{a_1}\plus{}\frac{2^2}{a_2}\plus{}\ldots \plus{}\frac{n^2}{a_n}\geq \frac{n\plus{}2}{2}\minus{}\frac{1}{n^2\plus{}1}

Problem Statement