1971 IMO 第 1 题

(BUL 2) Consider a sequence of polynomials $P_{0}(x), P_{1}(x), P_{2}(x), \ldots$, $P_{n}(x), \ldots$, where $P_{0}(x)=2, P_{1}(x)=x$ and for every $n \geq 1$ the following equality holds: $$P_{n+1}(x)+P_{n-1}(x)=x P_{n}(x)$$ Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1$, $$\left(x^{2}-4\right)\left[P_{n}^{2}(x)-4\right]=\left[a P_{n+1}(x)+b P_{n}(x)+c P_{n-1}(x)\right]^{2}$$

(BUL 2) 考虑一系列多项式 $P_{0}(x), P_{1}(x), P_{2}(x), \ldots$, $P_{n}(x), \ldots$，其中 $P_{0}(x)=2, P_{1}(x)=x$ 并且对于每个 $n \geq 1$ 存在以下等式： $$P_{n+1}(x)+P_{n-1}(x)=x P_{n}(x)$$证明存在三个实数$a, b, c$，使得对于所有$n \geq 1$, $$\left(x^{2}-4\right)\left[P_{n}^{2}(x)-4\right]=\left[a P_{n+1}(x)+b P_{n}(x)+c P_{n-1}(x)\right]^{2}$$