1986 AMC 12 第 30 题
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30.
联立方程组 的实数解 的个数是
The number of real solutions of the simultaneous equations is
答案：B
小提示：
这些方程迫使四个变量同号
The equations force all four variables to have the same sign
大提示：
对正数 ，研究 与 以及 的大小关系
For positive study relative to and to
解答：
每个表达式 都与 同号，所以四个变量必定同号。先假设它们全为正数。由算术平均值与几何平均值不等式，每个变量都至少为 。当 时， 如果任一变量大于 ，这些方程就会导出不可能成立的严格循环不等式 。所以唯一的正数解是 将四个变量同时取相反数后方程组不变，因此恰好还有一个负数解，其中所有变量都等于 。所以共有两个实数解。
因此正确答案是 B。
Each expression has the same sign as so all four variables have the same sign. Suppose first that they are positive. By AM-GM, every variable is at least For If any variable exceeded the equations would give the impossible strict cycle Hence the only positive solution is Negating all four variables preserves the system, giving exactly one negative solution, with all variables equal to Thus there are two real solutions.
Therefore the correct answer is B.
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