2015 AMC 10B Problem 14

Question

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

\begin{aligned} &(x-a)(x-b) \\ &\quad +(x-b)(x-c)=0? \end{aligned}

15

15.5

16

16.5

17

Factor out

x-b

Maximize

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

using distinct one-digit numbers

Factoring the left-hand side gives

(x-b)(2x-a-c)=0.

Thus the roots are

b

and

\frac{a+c}{2},

whose sum is

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

The coefficient of

b

in this sum is twice the coefficient of either

a

or

c,

so assign the largest digit to

b.

The next two largest distinct digits should be

a

and

c.

Taking

b=9

and

\{a,c\}=\{7,8\}

gives

9+\frac{7+8}{2}=16.5.

Thus, the correct answer is D .

Problem 14 in other years