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Turkey NMO 2000 1st Round - P08 (Algebra)

Source:

July 24, 2012
function

Problem Statement

(x+y)5=z(y+z)5=x(z+x)5=y\begin{array}{rcl} (x+y)^5 &=& z \\ (y+z)^5 &=& x \\ (z+x)^5 &=& y \end{array} How many real triples (x,y,z)(x,y,z) are there satisfying above equation system?
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> Infinitely many<spanclass=′latex−bold′>(E)</span> None <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ \text{Infinitely many} \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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