Question

x+y+z=1, x/(1+x)+y/(1+y)+z/(1+z)<=3/4

x+y+z=1, x/(1+x)+y/(1+y)+z/(1+z)<=3/4都是正数

Answer 1

x/(1+x)+y/(1+y)+z/(1+z)≤3/4
即1-1/(1+x)+1-1/(1+y)+1-1/(1+z)≤3/4
即1/(1+x)+1/(1+y)+1/(1+z)≥9/4
即4[1/(1+x)+1/(1+y)+1/(1+z)]≥9
即[(1+x)+(1+y)+(1+z)][1/(1+x)+1/(1+y)+1/(1+z)]≥9
由于(1+x)+(1+y)+(1+z)≥3倍的3次根号下[(1+x)(1+y)(1+z)]
1/(1+x)+1/(1+y)+1/(1+z)≥3倍的3次根号下[1/(1+x)(1+y)(1+z)]
所以[(1+x)+(1+y)+(1+z)][1/(1+x)+1/(1+y)+1/(1+z)]≥9
所以原不等式成立.