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2012 AMC 12A Problem 6

Problem 6 of 25EasierAlgebra

The sums of three whole numbers taken in pairs are 12,12, 17,17, and 19.19. What is the middle number?

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Solution

Let the numbers be a<b<c.a \lt b \lt c. Adding the three pairwise sums gives 2(a+b+c)=12+17+192(a+b+c) = 12+17+19 =48,= 48, so a+b+c=24.a+b+c = 24. Then a=24−19=5,a = 24-19 = 5, b=24−17=7,b = 24-17 = 7, and c=24−12=12.c = 24-12 = 12. The middle number is 7.7. Thus, the correct answer is D.

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Concepts: system of equations

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.