**A. 434.** Points *A*, *B*, *C* lie inside the convex hexagon *MNPQRS*, such that the triangles *ABC*, *NAM*, *PQB* and *CRS* are similar. Let *X*, *Y*, *Z* be the midpoints of the line segments *NP*, *QR*, *SM*, respectively, and let *G*, *K*, *I* be the centroids of the triangles *ABC*, *MPR* and *NQS*, respectively. Prove that (*a*) If triangle *ABC* is equilateral then triangle *GKI* is equilateral; (*b*) Triangles *ABC* and *XYZ* are similar if and only if triangle *ABC* is equilateral.

*Romanian competition problem*

(5 points)