1990 CMO 第 4 题

Given a positive integer number $a$ and two real numbers $A$ and $B$ , find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution:

$$\left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2 x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right.$$

给定一个正整数 $a$ 和两个实数 $A$ 和 $B$ ，找到 $A$ 和 $B$ 上下列方程组有整数解的充分必要条件：

\left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2 x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right。