Question

已知 x,y,z都是正实数,且 x+y+z=xyz 证明 (y+x)/z+(y+z)/x+(z+x)/y≥2(1/x+1

Answer 1

1／x＝p

1／y＝q

1／z＝r

pq+qr+pr=1

(y+x)/z+(y+z)/x+(z+x)/y≥2(1/x+1/y+1/z)^2

为(pq+qr+pr)[r/p+r/q+q/r+q/p+p/r+p/q]>=2(p+q+r)^2

即2(r^2+p^2+q^2+pq+qr+rp)+rrq/p+rrp/q+qqr/p+qqp/r+ppr/q+ppq/r>=2(p+q+r)^2

即rrq/p+rrp/q+qqr/p+qqp/r+ppr/q+ppq/r>=2(pq+qr+pr)

又因为rrp／q＋rrq/p>=2rr

所以只需证rr+pp+qq>＝pq+qr+pr

(r-p)^2+(p-q)^2+(q-r)^2>=0