Find the least positive integer such that there are at least 1000 unordered pairs of diagonals in regular polygon with vertices that intersect at right angle in the interior of the polygon.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. Let = 4k.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. (1 + 3 + 5 + \dots(2k − 1) + \dots3 + 1) k.
Hint 3: Proceed with the final algebraic steps to solve the system. = (k 2 + (k − 1)2 )k 1000.
Step 1: Let = 4k
Step 2: (1 + 3 + 5 + \dots(2k − 1) + \dots3 + 1) k
Step 3: = (k 2 + (k − 1)2 )k 1000
Step 4: => (2k 2 − 2k + 1) 1000
Step 5: => 9 as in N
Step 6: => 36
Step 7: (1 + 3 + \dots2k − 1) \times 2 \times (2k + 1)
Step 8: => (2k + 1) 2k 2 1000
Step 9: => 2 (2k + 1) 500
Step 10: => 30
Step 11: => min(36, 30) = 30
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