2014 AMC 12B Problem 10

Question

Danica drove her new car on a trip for a whole number of hours, averaging

55

miles per hour. At the beginning of the trip,

abc

miles was displayed on the odometer, where

abc

is a

3

-digit number with

a \ge 1

and

a+b+c \le 7.

At the end of the trip, the odometer showed

cba

miles. What is

a^2 + b^2 + c^2?

26

27

36

37

41

The distance driven is

cba - abc = 99(c-a)

It is also a multiple of

55,

so it is a multiple of

\operatorname{lcm}(99,55)=495

The distance driven is

cba - abc = 99(c-a),

a multiple of

9.

Driving a whole number of hours at

55

mph makes it a multiple of

55

too, hence a multiple of

495.

Since the odometer difference is at most a

3

-digit number and

a \ge 1,

the distance must be

495,

so

c - a = 5.

With

a \ge 1

and

a+b+c \le 7,

the only choice is

a=1,

c=6,

b=0.

Then

a^2+b^2+c^2 = 1 + 0 + 36 = 37.

Thus, the correct answer is D.

Problem 10 in other years