BG-MO 1966

(a) In the plane of the triangle $ABC$ , find a point with the following property: its symmetrical points with respect to the midpoints of the sides of the triangle lie on the circumscribed circle. (b) Construct the triangle $ABC$ if it is known the positions of the orthocenter $H$ , midpoint of the side $AB$ and the midpoint of the segment joining the feet of the heights through vertices $A$ and $B$ .

It is given a tetrahedron with vertices $A,B,C,D$ .

(a) Prove that there exists a vertex of the tetrahedron with the following property: the three edges of that tetrahedron through that vertex can form a triangle. (b) On the edges $DA,DB$ and $DC$ there are given the points $M,N$ and $P$ for which: $$DM=\frac{DA}n,\enspace DN=\frac{DB}{n+1}\enspace DP=\frac{DC}{n+2}$$ where $n$ is a natural number. The plane defined by the points $M,N$ and $P$ is $\alpha_n$ . Prove that all planes $\alpha_n$ , $(n=1,2,3,\ldots)$ pass through a single straight line.