2007 IMO Shortlist A6

Let $a_{1}, a_{2}, \ldots, a_{100}$ be nonnegative real numbers such that $a_{1}^{2}+a_{2}^{2}+\ldots+a_{100}^{2}=1$. Prove that $$a_{1}^{2} a_{2}+a_{2}^{2} a_{3}+\ldots+a_{100}^{2} a_{1}<\frac{12}{25}$$ (Poland)

令 $a_{1}, a_{2}, \ldots, a_{100}$ 为非负实数，使得 $a_{1}^{2}+a_{2}^{2}+\ldots+a_{100}^{2}=1$。证明 $$a_{1}^{2} a_{2}+a_{2}^{2} a_{3}+\ldots+a_{100}^{2} a_{1}<\frac{12}{25}$$(波兰)