A counterexample is $c_1= 2,c_2= 2,c_3= 81,x_1= -8,x_2= -1,x_3= \frac{2}{9},y_1= 0,y_2= 0,y_3= 0$, $z_1= 0,z_2= 0,z_3= 0$.
Iosif Pinelis
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