2010 AMC 12A Problems/Problem 13
Problem 13
For how many integer values of do the graphs of and not intersect?
Solution
We can see that the function is symmetric to the line , and the distance to the origin approaches infinity as the function approaches either the -axis or the -axis. Therefore, assuming that graphs don't intersect, the point at which the function is closest to the function (which is clearly a circle) is when . It follows that at these points, the magnitude of the and values for the function will be .
All of these points are found at angles , , , or , so the minimum distance from the origin to the function is .
The distance from the circle to the origin is always . Therefore, we want to find all integer values such that
$|k| < \sqrt{|2k|}}$ (Error compiling LaTeX. Unknown error_msg)
It is then easy to see that the only values that satisfy the inequality are and , a total of values.