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How many lattice points (points with both coordinates as integers) lie strictly inside the triangle with vertices at (0,0), (0,41), and (41,0)?

1. 820
2. 780
3. 901
4. 861

Correct answer: 780

Solution

Interior lattice points satisfy i >= 1, j >= 1, i + j <= 40. For each i from 1 to 39, j ranges from 1 to 40-i, giving (40-i) points. Total = sum_(i=1)³⁹(40-i) = sumₖ₌₁³⁹ k = 39*40/2 = 780.

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