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Philippine Mathematical Olympiad

Algebra Difficulty 4.3 AIME Find the answer Philippines

Problem:

How many pairs of integers solve the system $|x y| + |x - y| = 2$ if $-10 \leq x, y \leq 10$ ?

Your answer

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

Solution

Solution:

(ans. $4: (2, 0), (0, 2), (-2, 0), (0, -2)$ .
The cases are (A) $|x y| = 0$ and $|x - y| = 2 \Rightarrow (x, y) = (0, \pm 2), (\pm 2, 0)$ ; (B) $|x y| = 2$ and $|x - y| = 0 \Rightarrow$ no solution; (C) $|x y| = |x - y| = 1 \Rightarrow$ no solution.)

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.