RU-ORAL-GEOMETRY 2021

Points $A,B,C,D$ have been marked on checkered paper (see fig.). Find the tangent of the angle $ABD$ .

A trapezoid is given in which one base is twice as large as the other. Use one ruler (no divisions) to draw the midline of this trapezoid.

$ABCD$ is a convex quadrilateral such that $\angle A = \angle C < 90^{\circ}$ and $\angle ABD = 90^{\circ}$ . $M$ is the midpoint of $AC$ . Prove that $MB$ is perpendicular to $CD$ .