Olympiad Maths Prep

Track / Stage 5 / 9 of 400 #609 of 2000

Problem 609

AIME late
Algebra Difficulty 5.0 Find the answer

5. For which integers aa and bb does the system of equations

{∣ax+by+2∣=1(3x−5y−2)(5x+y−22)(x+3y+4)=0 \left\{\begin{array}{c} |a x+b y+2|=1 \\ (3 x-5 y-2)(5 x+y-22)(x+3 y+4)=0 \end{array}\right.

have five solutions?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Answer: [a=1−b,b∈Z,b≠1a=3−b,b∈Z,b≠2a=3n+1,b=5n+2,n∈Z,n≠0a=3n,b=5n+1,n∈Z,n≠0\left[\begin{array}{l}a=1-b, b \in Z, b \neq 1 \\ a=3-b, b \in Z, b \neq 2 \\ a=3 n+1, b=5 n+2, n \in Z, n \neq 0 \\ a=3 n, b=5 n+1, n \in Z, n \neq 0\end{array}\right.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.