RO-TST 1992 Problem 9

Let $x, y$ be real numbers such that $1\le x^2-xy+y^2\le2$ . Show that: a) $\dfrac{2}{9}\le x^4+y^4\le 8$ ; b) $x^{2n}+y^{2n}\ge\dfrac{2}{3^n}$ , for all $n\ge3$ .

Laurențiu Panaitopol and Ioan Tomescu