A 5x5 board has 25 cells.The numbers $\{1,2,3,4,5\}$ are written on every row,every column and the two main diagonals. A 5x5 board has $25$ cells. The numbers $\{1,2,3,4,5\}$ are written on every row,every column and the two main diagonals without any repetition. If the sum of the numbers of the diagonal below the leading diagonal be represented as $A$. Show that $A\ne 20$. Also find the maximum possible value of $A$.  
I could prove that $A\ne 20$ by showing that each of the cells of the diagonal below the leading diagonal cannot be $20$ as if such arrangement exists, then the board cannot have $5$ five times, which actually should be by the given condition of the question.  
For the second part, I can imagine the maxiumum value to be $17$. For which I proved that $5$ can occur only $2$ times in the four cells and probably $4$ cannot occur twice. But I got no formal proof. Please help!
 A: There can be at most $2$ equal numbers on subdiagonal:
$$
\begin{array}{|c|c|c|c|c|}
\hline
.&.&.&.&. \\
\hline
A&.&.&.&.\\
\hline
.&A&.&.&.\\
\hline
.&.&.&.&.\\
\hline
.&.&.&.&. \\ \hline
\end{array}
\implies
\begin{array}{|c|c|c|c|c|}
\hline
.&.&.&.&A \\
\hline
A&.&.&.&.\\
\hline
.&A&.&.&.\\
\hline
.&.&.&A&.\\
\hline
.&.&A&.&. \\ \hline
\end{array};
$$
$$
\begin{array}{|c|c|c|c|c|}
\hline
.&.&.&.&. \\
\hline
A&.&.&.&.\\
\hline
.&.&.&.&.\\
\hline
.&.&A&.&.\\
\hline
.&.&.&.&. \\ \hline
\end{array}
\implies
\varnothing;
$$
$$
\begin{array}{|c|c|c|c|c|}
\hline
.&.&.&.&. \\
\hline
A&.&.&.&.\\
\hline
.&.&.&.&.\\
\hline
.&.&.&.&.\\
\hline
.&.&.&A&. \\ \hline
\end{array}
\implies
\begin{array}{|c|c|c|c|c|}
\hline
.&A&.&.&. \\
\hline
A&.&.&.&.\\
\hline
.&.&A&.&.\\
\hline
.&.&.&.&A\\
\hline
.&.&.&A&. \\ \hline
\end{array};
$$
$$
\begin{array}{|c|c|c|c|c|}
\hline
.&.&.&.&. \\
\hline
.&.&.&.&.\\
\hline
.&A&.&.&.\\
\hline
.&.&A&.&.\\
\hline
.&.&.&.&. \\ \hline
\end{array}
\implies
\varnothing;
$$
(other cases are symmetrical).
Other $2$ cells cannot be filled with $2$ equal numbers, because we will get collision in red cell:
$$
\begin{array}{|c|c|c|c|c|}
\hline
.&.&.&.&. \\
\hline
A&.&.&.&.\\
\hline
.&A&.&.&.\\
\hline
.&.&B&.&.\\
\hline
.&.&.&B&. \\ \hline
\end{array}
\implies
\begin{array}{|c|c|c|c|c|}
\hline
.&.&.&.&\color{red}{A,B} \\
\hline
A&B&.&.&.\\
\hline
B&A&.&.&.\\
\hline
.&.&B&A&.\\
\hline
.&.&A&B&. \\ \hline
\end{array}.
$$
So, subdiagonal has either $4$ different numbers, or $2$ equal and $2$ different.
$4$ different numbers can reach only value of $14$: $5+4+3+2$.
$2$ equal and $2$ different numbers can reach value of $17$: $5+5+4+3$, and no more.
Examples:
$4\;3\;2\;1\;5 ~~~~~~~ 1\;5\;4\;3\;2$ 
$5\;2\;1\;4\;3 ~~~~~~~ 5\;3\;2\;4\;1$ 
$1\;5\;3\;2\;4 ~~~~~~~ 2\;4\;5\;1\;3$ 
$3\;1\;4\;5\;2 ~~~~~~~ 4\;1\;3\;2\;5$ 
$2\;4\;5\;3\;1 ~~~~~~~ 3\;2\;1\;5\;4$ 
