Two non-congruent triangles have the same area. Each triangle has sides of length 8 and 9, and the third side of each triangle has integer length. What is the sum of the lengths of the third sides?
WLOG, a≤a′.a2≤145⇒a≤12 By Triangle Inequality,a+8>9⇒a>1⇒a≥2a′2=290−a2=290−4,290−9,290−16,290−25,290−36,290−49,290−64,290−81,290−100,290−121,290−144⇒a′2=286×281×274×265×254×241×226×209×190×169✓146×⇒a′=13,a=11Ans=a+a′=11+13=24
