Sum of the entries
Posted: Thu Oct 13, 2016 6:01 pm
You are given a $n \times n$ array as follows:
\[\begin{array}{cccc}\displaystyle
1 & \frac 1 2 & \cdots & \frac 1 n\\
\frac 1 2 & \frac 1 3 & \cdots & \frac 1 {n+1}\\
: & : & & :\\
: & : & & :\\
\frac 1 n & \frac 1 {n+1} & \cdots & \frac 1 {2n-1}
\end{array}
\]
Prove that the sum of any $n$ entries situated in different rows and different columns is not less than $1$.
\[\begin{array}{cccc}\displaystyle
1 & \frac 1 2 & \cdots & \frac 1 n\\
\frac 1 2 & \frac 1 3 & \cdots & \frac 1 {n+1}\\
: & : & & :\\
: & : & & :\\
\frac 1 n & \frac 1 {n+1} & \cdots & \frac 1 {2n-1}
\end{array}
\]
Prove that the sum of any $n$ entries situated in different rows and different columns is not less than $1$.