Question 1199136
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<pre>

Your first idea is right.


    +-----------------------------------------------------------------+
    |    But in such problems, you ALWAYS must make one more (next)   |
    |    step forward and to check,  if integer solution does exist.  |
    +-----------------------------------------------------------------+


In case of  65  diagonals it does exist: the number of sides is n = 13.


In case of  " 80  diagonals "  integer solution  DOES  NOT  exist.


You will easily detect it, when you apply the quadratic formula.
</pre>


Not for every given &nbsp;" number of diagonals "  &nbsp;the integer solution does exist.



So, &nbsp;very often such problems have a &nbsp;{{{highlight(highlight(HUGE))}}} &nbsp;hidden underwater stone &nbsp;(as a trap),

and the duty of a person who solves such problem is to detect it / (to recognize it),

if such trap does present in the hidden form in the problem.



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By the way, &nbsp;the formulation of the problem starts with the words 


<pre>
    If a polygonal n sides has  (n/2)(n - 3)  diagonals . . . 
</pre>

The word  &nbsp;" if " &nbsp;is excessive in this phrase : &nbsp;the number of diagonals 

of any convex &nbsp;n-gon &nbsp;is  &nbsp;&nbsp;{{{(n*(n-3))/2}}},  &nbsp;&nbsp;for any &nbsp;n > 3, &nbsp;and without any  &nbsp;" if&nbsp;".