求证(x1+x2+...xn)^2/2(x1^2+x2^2+.xn^2)≤x1/(x2+x3)+x2/(x3+x4)+.xn/(x1+x2)
问题描述:
求证(x1+x2+...xn)^2/2(x1^2+x2^2+.xn^2)≤x1/(x2+x3)+x2/(x3+x4)+.xn/(x1+x2)
答
证明: 由排序不等式, x1^2+x2^2+...+xn^2>=x1x2+x2x3+...xn-1xn+xnx1 x1^2+x2^2+...+xn^2>=x1x3+x2x4+...xn-1x1+xnx2 两式相加得 2(x1^2+x2^2+...+xn^2)>=x1(x2+x3)+x2(x3+x4)+...+xn-1(xn+x1)+xn(x1+x2) 又因为由...