Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.1 AIME Find the answer United States

Problem:
How many lattice points are enclosed by the triangle with vertices $(0,99)$ , $(5,100)$ , and $(2003, 500)$ ? Don't count boundary points.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Using the determinant formula, we get that the area of the triangle is
$\left|\begin{array}{cc} 5 & 1 \\ 2003 & 401 \end{array}\right| / 2 = 1$
There are 4 lattice points on the boundary of the triangle (the three vertices and $(1004, 300)$ ), so it follows from Pick's Theorem that there are 0 in the interior.

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