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2024 AMC 10A Problem 23

Below is the professionally curated solution for Problem 23 of the 2024 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 10A solutions, or check the answer key.

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Concepts:system of equationsfactoringDiophantine Equation

Difficulty rating: 2270

23.

Integers a,a, b,b, and cc satisfy

ab+c=100,bc+a=87,ca+b=60.ab + c = 100, \quad bc + a = 87, \quad ca + b = 60.

What is ab+bc+ca?ab + bc + ca?

212212

247247

258258

276276

284284

Solution:

Add the three equations: (ab+bc+ca)+(a+b+c)=247.(ab + bc + ca) + (a + b + c) = 247. Now subtract them in pairs, which factors nicely as (a−c)(b−1)=13,(a - c)(b - 1) = 13, (b−a)(c−1)=27,(b - a)(c - 1) = 27, and (b−c)(a−1)=40.(b - c)(a - 1) = 40. These pin down (a,b,c)=(−9,−12,−8),(a, b, c) = (-9, -12, -8), so a+b+c=−29.a + b + c = -29. Then ab+bc+ca=247−(−29)=276.ab + bc + ca = 247 - (-29) = 276. Thus, D is the correct answer.

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