IMC 1995 Problem 6

Let $p>1$ . Show that there exists a constant $K_{p} >0$ such that for every $x,y\in \mathbb{R}$ with $|x|^{p}+|y|^{p}=2$ , we have $$(x-y)^{2} \leq K_{p}(4-(x+y)^{2}).$$