Prove that the equation $$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.
1966 Bulgaria National Olympiad
Prove that for every four positive numbers $a,b,c,d$ the following inequality is true: $$\sqrt{\frac{a^2+b^2+c^2+d^2}4}\ge\sqrt[3]{\frac{abc+abd+acd+bcd}4}.$$
(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.