KURSCHAK 1976

$ABCD$ is a parallelogram. $P$ is a point outside the parallelogram such that angles $\angle PAB$ and $\angle PCB$ have the same value but opposite orientation. Show that $\angle APB = \angle DPC$ .

A lottery ticket is a choice of $5$ distinct numbers from $1, 2,3,...,90$ . Suppose that $5^5$ distinct lottery tickets are such that any two of them have a common number. Prove that one can find four numbers such that every ticket contains at least one of the four.

Prove that if the quadratic $x^2 +ax+b$ is always positive (for all real $x$ ) then it can be written as the quotient of two polynomials whose coefficients are all positive.