<prompt> Tafuta idadi ya nambari za kweli zilizoagizwa mara tatu x,y,z kama hizokushoto x frac1y kulia kushoto y frac1z kulia kushoto z frac1x kulia 8.  <response>: Na AM-GM,anzaalignx frac1y ge 2 sqrtfracxy, y frac1z ge 2 sqrtfracyz, z frac1x ge 2 sqrtfraczx,mwishoalignhivyokushoto x frac1y kulia kushoto y frac1z kulia kushoto z frac1x kulia ge 2 sqrtfracxy cdot 2 sqrtfracyz cdot 2 sqrtfraczx 8. Usawa hutokea wakati x frac1y, y frac1z, na z frac1x, au xy 1, yz 1, na xz 1. Tukizidisha milinganyo yote mitatu, tunapata x2 y2 z2 1. Kwa kuwa x, y, na z zote ni chanya. ,xyz 1.Kugawa yz 1, tunapata x 1. Vile vile, y 1 na z 1, kwa hivyo kuna 1 triple x,y pekee, z, yaani 1,1,1. 