KURSCHAK 1978

$a$ and $b$ are rationals. Show that if $ax^2 + by^2 = 1$ has a rational solution (in $x$ and $y$ ), then it must have infinitely many.

The vertices of a convex $n$ -gon are colored so that adjacent vertices have different colors. Prove that if $n$ is odd, then the polygon can be divided into triangles with non-intersecting diagonals such that no diagonal has its endpoints the same color.

A triangle has inradius $r$ and circumradius $R$ . Its longest altitude has length $H$ . Show that if the triangle does not have an obtuse angle, then $H \ge r+R$ . When does equality hold?