bài 4:
Đặt : 1x2−1=8a3,1y2−1=8b3,1z2−1=8c3⇒abc=1,(a,b,c>0)
Ta có : 1x2=8a3+1=(2a+1)(4a2−2a+1)≤(2a+1+4a2−2a+12)2=(2a2+1)2
Tương tự suy ra :x+y+z≥12a2+1+12b2+1+12c2+1
Ta CM : T=12a2+1+12b2+1+12c2+1≥1
Giả sử c bé nhất ⇒0<c≤1⇒ab≥1
Khi đó : 12a2+1+12b2+1=1−4a2b2−1(2a2+1)(2b2+1)≥1−4a2b2−1(2ab+1)2=22ab+1
Suy ra : T≥22ab+1+12c2+1=2cc+2+12c2+1=1+2c(c−1)2(c+2)(2c2+1)≥1,(đpcm)