1987 Putnam A4

Let $P$ be a polynomial, with real coefficients, in three variables

and $F$ be a function of two variables such that

$$

P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \text{for all real $x,y,z,u$},

$$

and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let

$A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.

令 $P$ 为具有实数系数、三个变量的多项式

和 $F$ 是两个变量的函数，使得

$$

P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \text{对于所有实数 $x,y,z,u$},

$$

使得 $P(1,0,0)=4$，$P(0,1,0)=5$，$P(0,0,1)=6$。还让

$A,B,C$ 为复数，$P(A,B,C)=0$ 且 $|B-A|=10$。查找$|C-A|$。