1968 IMO 第 4 题

(BUL 2) ${ }^{\mathrm{IMO} 3}$ Let $a, b, c$ be real numbers. Prove that the system of equations $$\left\{\begin{array}{r} a x_{1}^{2}+b x_{1}+c=x_{2} \\ a x_{2}^{2}+b x_{2}+c=x_{3} \\ \cdots \cdots \cdots \cdots \\ a x_{n-1}^{2}+b x_{n-1}+c=x_{n} \\ a x_{n}^{2}+b x_{n}+c=x_{1} \end{array}\right.$$ has a unique real solution if and only if $(b-1)^{2}-4 a c=0$. Remark. It is assumed that $a \neq 0$.

(BUL 2) ${ }^{\mathrm{IMO} 3}$ 令 $a, b, c$ 为实数。证明方程组 \left\{\begin{array}{r} a x_{1}^{2}+b x_{1}+c=x_{2} \\ a x_{2}^{2}+b x_{2}+c=x_{3} \\ \cdots \cdots \cdots \cdots \\ a x_{n-1}^{2}+b x_{n-1}+c=x_{n} \\ a x_{n}^{2}+b x_{n}+c=x_{1} \end{array}\right。 有唯一实数解当且仅当 $(b-1)^{2}-4 a c=0$。评论。 It is assumed that $a \neq 0$.