Find the least positive integer n such that there are at least 1000 unordered pairs of diagonals in a regular polygon with n vertices that intersect at a right angle in the interior of the polygon.

Step-by-step solution

Case-I: Let n=4 k aligned & (1+3+5+ (2 k-1)+ 3+1) k & = (k^2+(k-1)^2 ) k 1000 & k (2 k^2-2 k+1 ) 1000 & k 9 as k N & 4 k 36 & n 36 aligned Case-II: aligned & n=4 k+2 & (1+3+ 2 k-1) 2 (2 k+1) & (2 k+1) 2 k^2 1000 & k^2(2 k+1) 500 & k 7 & n 30 & (36,30)=30 aligned