OME 1969

Find the locus of the centers of the inversions that transform two points $A, B$ of a given circle $\gamma$ , at diametrically opposite points of the inverse circles of $\gamma$ .

Find the locus of the affix $M$ , of the complex number $z$ , so that it is aligned with the affixes of $i$ and $iz$ .

A bag contains plastic cubes of the same size, whose faces have been painted in colors: white, red, yellow, green, blue and violet (without repeating a color on two faces of the same cube). How many of these cubes can there be distinguishable to each other?

A circle of radius $R$ is divided into $8$ equal parts. The points of division are denoted successively by $A, B, C, D, E, F , G$ and $H$ . Find the area of the square formed by drawing the chords $AF$ , $BE$ , $CH$ and $DG$ .

Show that a convex polygon with more than four sides cannot be decomposed into two others, both similar to the first (directly or inversely), by means of a single rectilinear cut. Reasonably specify which are the quadrilaterals and triangles that admit a decomposition of this type.

Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities: $$P(x) \le P(x)^2; \,\, P(x) < 1 + P(x)^2; \,\, P(x) \le \frac12 +\frac12 P(x)^2.$$ Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$ , at the same time $-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$ .