CZ-SK-MO 1955

Consider a trapezoid $ABCD,AB\parallel CD,AB>CD.$ Let us denote intersections of lines as follows: $E=AC\cap BD, F=AD\cap BC.$ Let $GH$ be a line such that $G\in AD,H\in BC, E\in GH,GH\parallel AB.$ Moreover, denote $K,L$ midpoints of the bases $AB,CD$ respectively. Show that (a) the points $K,L$ lie on the line $EF,$ (b) lines $AC,KH$ and $BD,KG$ are not parallel (denote $M=AC\cap KH,N=BD\cap KG$ ), (c) the points $F,M,N$ are collinear.

Let $\mathsf{S}_1,\mathsf{S}_2$ be concentric spheres with radii $a,b$ respectively, where $a<b.$ Denote $ABCDA'B'C'D'$ a square cuboid ( $ABCD,A'B'C'D$ are the squares and $AA'\parallel BB'\parallel CC'\parallel DD'$ ) such that $A,B,C,D\in\mathsf{S}_2$ and the plane $A'B'C'D'$ is tangent to $\mathsf{S}_1.$ Finally assume that $$\frac{AB}{AA'}=\frac ab.$$ Compute the lengths $AB,AA'.$ How many of such cuboids exist (up to a congruence)?

In the complex plane consider the unit circle with the origin as its center. Furthermore, consider inscribed regular 17-gon with one of its vertices being $1+0i.$ How many of its vertices lie in the (open) unit disc centered in $\sqrt{3/2}(1+i)$ ?

Given that $a,b,c$ are distinct real numbers, show that the equation $$\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}=0$$ has a real root.