NMO Selection Tests for BMO and IMO

Since the problem is of an affine nature, we may (and will) assume that the triangle $XYZ$ is equilateral. The triangle $ABC$ has at least one vertex angle, say at $A$, greater than or equal to $60^\circ$, so $A$ is covered by the closed circumdisc $OYZ$, where $O$ is the center of the triangle $XYZ$. Since the latter is covered by the $4$-fold blow-up of the triangle $XYZ$ from $O$, the conclusion follows.

Suppose, if possible, that none of the vertices $A, B, C$ is covered by the $4$-fold blow-up of the triangle $XYZ$ from its centroid. Then the distance of the point $A$ to the line $YZ$ is greater than the distance of the point $X$ to this line, so the area of the triangle $AYZ$ is greater than the area of the triangle $XYZ$. Similarly, the triangles $BZX$ and $CXY$ both have an area greater than that of the triangle $XYZ$, in contradiction with the well known fact that of the four triangles $AYZ, BZX, CXY, XYZ$, the latter has not the smallest area.