HMMT February

Algebra Difficulty 4.9 AIME Find the answer

A triple of integers $(a, b, c)$ satisfies $a+b c=2017$ and $b+c a=8$ . Find all possible values of $c$ .

Your answer

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

Solution

Add and subtract the two equations to find $(b+a)(c+1)=8+2017$ and $(b-a)(c-1)=2017-8$ . We see that $c$ is even and then that every integer $c$ with $c+1|2025, c-1| 2009$ works. We factor and solve. The full solutions are $(2017,8,0),(-667,1342,2),(-59,-346,-6),(-31,256,8)$ .

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