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

$26$

$27$

$36$

$37$

$41$

Answer: D

Concepts:place value divisibility least common multiple

Difficulty rating: 1680

Solution:

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.

2014 AMC 12B Problem 10

Answer: D

Difficulty rating: 1680

Solution:

Problem 10 in Other Years