IMO 1976 LL VIE49

Determine whether there exist 1976 nonsimilar triangles with

IMO 1976 LL VIE49

Origin: VIE

Problem

Determine whether there exist 1976 nonsimilar triangles with angles \alpha, \beta, \gamma, each of them satisfying the relations sin \alpha + sin \beta + sin \gamma cos \alpha + cos \beta + cos \gamma = 12 and sin \alpha sin \beta sin \gamma = 12 25.