CZ-SK-MO 1956

Find all $x,y\in\left(0,\frac{\pi}{2}\right)$ such that \begin{align*} \frac{\cos x}{\cos y}&=2\cos^2 y, \frac{\sin x}{\sin y}&=2\sin^2 y. \end{align*}

In a given plane $\varrho$ consider a convex quadrilateral $ABCD$ and denote $E=AC\cap BD.$ Moreover, consider a point $V\notin\varrho$ . On rays $VA,VB,VC,VD$ find points $A',B',C',D'$ respectively such that $E,A',B',C',D'$ are coplanar and $A'B'C'D'$ is a parallelogram. Discuss conditions of solvability.

Find all real pairs $x,y$ such that \begin{align*} x-|y+1|&=1, x^2+y&=10. \end{align*}

Let a semicircle $AB$ be given and let $X$ be an inner point of the arc. Consider a point $Y$ on ray $XA$ such that $XY=XB$ . Find the locus of all points $Y$ when $X$ moves on the arc $AB$ (excluding the endpoints).