2022 AMC 12A Problems/Problem 13
Problem
Let be the region in the complex plane consisting of all complex numbers that can be written as the sum of complex numbers and , where lies on the segment with endpoints and , and has magnitude at most . What integer is closest to the area of ?
Solution
If is a complex number and , then the magnitude (length) of is . Therefore, has a magnitude of 5. If has a magnitude of at most one, that means for each point on the segment given by , the bounds of the region could be at most 1 away. Alone the line, excluding the endpoints, a rectangle with a width of 2 and a length of 5, the magnitude, would be formed. At the endpoints, two semicircles will be formed with a radius of 1 for a total area of . Therefore, the total area is .
~juicefruit