设ai>0(i=1,2,……n)且a1+a2+……+an=1,求证:a1^2/(a1+a2)+a2^2/(a2+a3)+……+an^2/(an+a1)大于等于1/2
问题描述:
设ai>0(i=1,2,……n)且a1+a2+……+an=1,求证:a1^2/(a1+a2)+a2^2/(a2+a3)+……+an^2/(an+a1)大于等于1/2
答
由柯西不等式(a1+a2+a2+a3+a3+a4+.+an+a1)*[a1^2/(a1+a2)+a2^2/(a2+a3)+……+an^2/(an+a1)]>=[√(a1^2)+√(a2^2)+...+√(an^2)]^2即2*[a1^2/(a1+a2)+a2^2/(a2+a3)+……+an^2/(an+a1)]>=(a1+a2+……+an)^2=1a1^2/(a1+...