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2013 AMC 10B Problem 5

Problem 5 of 25EasierAlgebra

Positive integers aa and bb are each less than 6.6. What is the smallest possible value of the following expression? 2⋅a−a⋅b2 \cdot a - a \cdot b

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Solution

The expression is 2a−ab=a(2−b)2a-ab=a(2-b). To make it as small as possible, choose bb as large as possible so that 2−b2-b is most negative, and choose aa as large as possible. Since a,b<6a,b<6, take a=b=5a=b=5. The minimum value is 5(2−5)=−155(2-5)=-15. Thus, the correct answer is B .

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Concepts: factoring · optimization

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