BG-MO 1968

Find all natural values of $k$ for which the system $$\begin{cases}x_1+x_2+\ldots+x_k=9\frac1{x_1}+\frac1{x_2}+\ldots+\frac1{x_k}=1\end{cases}$$ has solutions in positive numbers. Find these solutions.

I. Dimovski

On the line $g$ we are given the segment $AB$ and a point $C$ not on $AB$ . Prove that on $g$ , there exists at least one pair of points $P,Q$ symmetrical with respect to $C$ , which divide the segment $AB$ internally and externally in the same ratios, i.e $$\frac{PA}{PB}=\frac{QA}{QB} (1)$$ If $A,B,P,Q$ are such points from the line $g$ satisfying $(1)$ , prove that the midpoint $C$ of the segment $PQ$ is the external point for the segment $AB$ .

K. Petrov

The point $M$ is inside the tetrahedron $ABCD$ and the intersection points of the lines $AM,BM,CM$ and $DM$ with the opposite walls are denoted with $A_1,B_1,C_1,D_1$ respectively. It is given also that the ratios $\frac{MA}{MA_1}$ , $\frac{MB}{MB_1}$ , $\frac{MC}{MC_1}$ , and $\frac{MD}{MD_1}$ are equal to the same number $k$ . Find all possible values of $k$ .

K. Petrov

Find the kind of a triangle if $$\frac{a\cos\alpha+b\cos\beta+c\cos\gamma}{a\sin\alpha+b\sin\beta+c\sin\gamma}=\frac{2p}{9R}.$$ ( $\alpha,\beta,\gamma$ are the measures of the angles, $a,b,c$ are the respective lengths of the sides, $p$ the semiperimeter, $R$ is the circumradius)

K. Petrov