2015 AMC 12B Problem 12

Attempt Problem 12 of the 2015 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12B solutions, or check the answer key.

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12.

Let

a,

b,

and

c

be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation

(x - a)(x - b)

+ (x - b)(x - c) = 0?

15

15.5

16

16.5

17

Factor out the common

(x - b)

The roots are

b

and

\dfrac{a + c}{2};

maximize

b + \dfrac{a + c}{2}

over distinct digits

Factoring gives

(x - b)\bigl(2x - (a + c)\bigr) = 0,

so the roots are

b

and

\dfrac{a + c}{2}.

Their sum is

b + \dfrac{a + c}{2}.

Using distinct digits, take

b = 9

and

a + c = 8 + 7 = 15,

giving

9 + 7.5 = 16.5.

Thus, the correct answer is D.

Problem 12 in other years