NMC 1987 Problem 4

Let $a, b$, and $c$ be positive real numbers. Prove:

$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \leq \frac{a^{2}}{b^{2}}+\frac{b^{2}}{c^{2}}+\frac{c^{2}}{a^{2}}$$