2021 AIME II Problem 5

For positive real numbers s, let \tau(s) denote the set of all obtuse triangles that have area s and two sides with lengths 4 and 10. The set of all s for which \tau(s) is nonempty, but all triangles in \tau(s) are congruent, is an interval [a,b). Find a^2+b^2.

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.