Find all positive real numbers x,y,z such that 2 x − 2 y + 1 z = 1 2014 , 2 y − 2 z + 1 x = 1 2014 , 2 z − 2 x + 1 y = 1 2014

Question

Find all positive real numbers x,y,z such that  2  x  −  2  y  +      1    z    =      1    2014    ,    2  y  −  2  z  +      1    x    =      1    2014    ,    2  z  −  2  x  +      1    y    =      1    2014

Michaela Alvarado · Accepted Answer

Step 1Consider the system of equations  x  y  +  z  =  1  ,    y  z  +  x  =  1  ,    z  x  +  y  =  1These equations are related by cyclic permutations of (x, y, z), but they are satisified by (1,1,0) (and its cyclic permutations) when x, y and z are not all equal.There are also solutions where   x  =  y  =  z  =            ±              5            −      1        2   , but these are not the only solutions.

Carly Roy · Accepted Answer

Step 1However, in this case you can make a simple argument that starts with an observation based on cyclic invariance, namely that you may as well assume   x  ≥  y  ,  z (i.e., cycle through (x, y, z), (y, z, x) and (z, x, y) and pick the one that starts with the largest of the three numbers).If   x  ≥  y  ,  z , then   2  x  −  2  y  ≥  0 while   2  z  −  2  x  ≤  0 , so that            1      2014        =  2  x  −  2  y  +            1      z        ≥            1      z          ⟹    z  ≥  2014and            1      2014        =  2  z  −  2  x  +            1      y        ≤            1      y          ⟹    2014  ≥  yso we now have   x  ≥  z  ≥  2014  ≥  y . But this now tells us   2  y  −  2  z  ≤  0 , so that            1      2014        =  2  y  −  2  z  +            1      x        ≤            1      x          ⟹    2014  ≥  xso we now have   2014  ≥  x  ≥  z  ≥  2014  ≥  y , from which we see   x  =  z  =  2014  ≥  y . The final equality,   2014  =  y , comes by sustituting   x  =  z  =  2014 into any of the three equations.Note, the implication             1      2014        ≥            1      z          ⟹    z  ≥  2014 requires the assumption   z  &amp;gt;  0 .

Find all positive real numbers x,y,z such that 2 x − 2 y +

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