关于几个字母对称的等式或不等式证明题,在中学数学练习题和中学生数学竞赛题中是常见的。不少学生对解答它们感到困难,一是因为较繁,二是不知如何下手分析。其实,抓住对称这个特点,这两方面的困难都易于解决。一、对代表性的结构、形式和其他项间的关系进行分析,进而根据对称列出解答。例1.已知数a_1,a_2……,a_n满足a_1+a_2+……+a_n=1和a_1~2+a_2~2+……+a_n~2=1, 求证:S=a_1a_2+a_1a_3+……+a_(n-1)a_n=0(这里S代表所有可能的乘积 Symmetrical equations or inequality propositions on several letters are common in high school math practice and high school math competition problems. Many students find it difficult to answer them. First, they are more complicated, and second, they do not know how to start analysis. In fact, by seizing the symmetry feature, both difficulties are easy to solve. 1. Analyze representative structures, forms, and relationships among other items, and then list solutions based on symmetry. Example 1. Known numbers a_1, a_2,..., a_n satisfy a_1+a_2+...+a_n=1 and a_1~2+a_2~2+...+a_n~2=1, verification: S=a_1a_2+a_1a_3+... +a_(n-1)a_n=0 (where S represents all possible products