2024 AMC 12A Problem 17

Below is the professionally curated solution for Problem 17 of the 2024 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 12A solutions, or check the answer key.

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Concepts:system of equations factoring casework

Difficulty rating: 1890

17.

Integers $a,\ b,$ and $c$ satisfy $ab+c=100,$ $bc+a=87,$ and $ca+b=60.$ What is $ab+bc+ca?$

$212$

$247$

$258$

$276$

$284$

Solution:

Subtracting pairs gives $(a-c)(b-1)=13,\ (a-b)(c-1)=-27,$ and $(b-c)(a-1)=40.$ Because $13$ is prime, only a few cases arise; testing them yields $a=-9,\ b=-12,\ c=-8,$ which satisfy all three original equations. Then $ab+bc+ca=108+96+72=276.$ Thus, the correct answer is D.

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