IR-MO 2006

We have finite number of distinct shapes in plane. A "convex Kearting" of these shapes is covering plane with convex sets, that each set consists exactly one of the shapes, and sets intersect at most in border. In which case Convex kearting is possible?

1. Finite distinct points
2. Finite distinct segments
3. Finite distinct circles

We mean a traingle in $\mathbb Q^{n}$ , 3 points that are not collinear in $\mathbb Q^{n}$

a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$ . Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$ . b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^{n}$ it can be embedded in $\mathbb Q^{m}$ . c) Find a triangle that can be embedded in $\mathbb Q^{n}$ and no triangle similar to it can be embedded in $\mathbb Q^{3}$ . d) Find a natural $m'$ that for each traingle that can be embedded in $\mathbb Q^{n}$ then there is a triangle similar to it, that can be embedded in $\mathbb Q^{m}$ . You must prove the problem for $m=9$ and $m'=6$ to get complete mark. (Better results leads to additional mark.)