Problem:
For positive real numbers , let denote the set of all obtuse triangles that have area and two sides with lengths and . The set of all for which is nonempty, but all triangles in are congruent, is an interval . Find .
Solution:
Answer (736):
Suppose that triangles and have the same area, both have sides with lengths 4 and 10, and the angles between the sides of lengths 4 and 10 in and are and , respectively, with . Let be the other angle in adjacent to its side of length 4 . Because the common area of the triangles is , it follows that and are supplementary, so the two triangles can be placed side by side along their sides of length 4 so that their sides of length 10 lie along a straight line, as shown.
Exactly one of these triangles is obtuse if and . The common area of the triangles is an increasing function of . When , the legs of the right triangle are 4 and , so the common area of the triangles is . When , the common area is . The required interval is . The requested value is .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions.