AIME I, 2025

Algebra Difficulty 4.2 AIME Find the answer United States

Find the number of ordered pairs $(x, y)$ , where both $x$ and $y$ are integers between $-100$ and $100$ , inclusive, such that $12x^2 - xy - 6y^2 = 0$ .

Your answer

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

Solution

The given equation can be written as $(3x + 2y)(4x - 3y) = 0$ . Thus the graph of this equation consists of two lines intersecting at the origin. Lattice points (points with integer coordinates) on this graph that make the first factor equal to $0$ are of the form $(2k, -3k)$ for some integer $k$ . In order for the coordinates to be in the required range, it must be that $-33 \le k \le 33$ , giving $67$ lattice points. Lattice points that make the second factor equal to $0$ are of the form $(3\ell, 4\ell)$ for some integer $\ell$ . In order for the coordinates to be in the required range, it must be that $-25 \le \ell \le 25$ , giving $51$ lattice points. The lattice point $(0, 0)$ occurs in each case. The requested number of points is therefore $67 + 51 - 1 = 117$ .

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