2026 AIME II Problem 6

Find the sum of all real numbers r such that there is at least one point where the circle with radius r centered at (4, 39) is tangent to the parabola with equation 2y = x^2 - 8x + 12.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.