Question

What is the sum of all integer solutions to $1 < (x-2)^2 < 25?$

$10$

$12$

$15$

$19$

$25$

Answer: B

Concepts:inequality absolute value symmetry

Difficulty rating: 1140

Solution:

Suppose we have $x=2+k$ as a solution. Then, $x=2-k$ would also be a solution as $((2+k)-2)^2 = ((2-k)-2)^2$ The sum of these two solutions would be $4.$ Thus, the sum of all integer solutions to the above equation is four times the number of positive $k$ 's that work.

To find the number of $k$ 's, we need to find the number of positive solutions to: $1 < k^2 < 25,$ which would be $3,$ as $k=2,3,4.$

Therefore, the sum of all the solutions is $4\cdot3=12.$

Thus, the correct answer is B .

2012 AMC 10B Problem 8

Answer: B

Difficulty rating: 1140

Solution:

Problem 8 in Other Years