CZ-SK-MO 1952

Let $a,b,c$ be positive rational numbers such that $\sqrt a+\sqrt b=c$ . Show that $\sqrt a$ and $\sqrt b$ are also rational.

Let $ABCD$ be a convex quadrilateral with $AB=CD$ . Let $R,S$ be midpoints of sides $AD,BC$ respectively. Consider rays $AU, DV$ parallel with ray $RS$ and all of them point in the same direction. Show that $\angle BAU=\angle CDV$ .

Let $p,q$ be positive integers. Consider a rectangle $ABCD$ with lengths of sides $p$ and $q$ that consists of $pq$ unital squares. How many of these squares are crossed by diagonal $AC$ ?