设a,b,c为正数,求证c/(a+b)+b/(c+a)+a/(b+c)>=3如题

Question

设a,b,c为正数,求证c/(a+b)+b/(c+a)+a/(b+c)>=3
如题

Answer 1

题目是：设a,b,c为正数,求证c/(a+b)+b/(c+a)+a/(b+c)>=3/2 a/(b+c)=(a+b)/2(b+c)+(a+c)/2(b+c)-1/2 b/(a+c)=(a+b)/2(a+c)+(b+c)/2(a+c)-1/2 c/(a+b)=(c+a)/2(a+b)+(b+c)/2(a+b)-1/2 三个式子相加: a/(b+c)+b/(a+c)+c/(a+b)=[(a+b)/2(b+c)+(b+c)/2(a+b)]+[(a+c)/2(b+c)+(b+c)/2(a+c)]+[(a+b)/2(a+c)+(a+c)/2(a+b)]-3/2 >=2*1/2+2*1/2+2*1/2-3/2 =3/2 即：a/(b+c)+b/(c+a)+c/(a+b) >=3/2