CZ-SK-MO 1954

Solve the equation $$ax^2+2(a-1)x+a-5=0$$ in real numbers with respect to (real) parametr $a$ .

Let $a,b$ complex numbers. Show that if the roots of the equation $z^2+az+b=0$ and 0 form a triangle with the right angle at the origin, then $a^2=2b\neq0.$ Also determine whether the opposite implication holds.

Show that $$\log_2\pi+\log_4\pi<\frac52.$$

Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$ ). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$ . (a) Let $M=B'Y\cap D'X$ . Find the locus of all $M$ s. (b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.