OSN 2014 Problem 8

A positive integer is called beautiful if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$ . A positive integer that is not beautiful is ugly.

a) Prove that $2014$ is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.