Question

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

Answer

Answer: 117

Concepts:Diophantine Equation factoring counting integers in a range

Difficulty rating: 2110

Solution:

The equation factors as $\begin{gathered} 12x^2 - xy - 6y^2 \\ = (3x + 2y)(4x - 3y) \\ = 0, \end{gathered}$ so every solution has $4x = 3y$ or $3x = -2y.$

Integer solutions of $4x = 3y$ are $(x, y) = (3t, 4t);$ the constraint $|4t| \le 100$ gives $-25 \le t \le 25,$ or $51$ pairs. Integer solutions of $3x = -2y$ are $(x, y) = (2t, -3t);$ the constraint $|3t| \le 100$ gives $-33 \le t \le 33,$ or $67$ pairs. The families overlap only at $(0, 0),$ so the count is $51 + 67 - 1 = 117.$

2025 AIME I Problem 4

Answer: 117

Difficulty rating: 2110

Solution:

Problem 4 in Other Years