CZ-SK-MO 1957

Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$ . Then, solve the equation for the determined values of $p$ .

Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$ . Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$ . (1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$ , where $X$ lies on line $VA$ and $Y$ lies on line $BC$ . Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$ . (2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$

Find all real numbers $\alpha$ such that both values $\cot(\alpha)$ and $\cot(2\alpha)$ are integers.

Consider a non-zero convex angle $\angle POQ$ and its inner point $M$ . Moreover, let $m>0$ be given. Construct a trapezoid $ABCD$ satisfying the following conditions: (1) vertices $A, D$ lie on ray $OP$ and vertices $B,C$ lie on ray $OQ$ , (2) diagonals $AC$ and $BD$ intersect in $M$ , (3) $AB=m$ . Prove that your construction is correct and discuss conditions of solvability.