Diophantine equation : "(xy + yz + zx) / (x + y + z)=4

hvp001hvp

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Find positive integer solutions (x,y,z) that satisfy the following equation: (xy+yz+zx)/(x+y+z)=4.Thanks for help.
 
IMHO this problem is more like one of those challenging Olympiad-style questions than homework. Here is an idea.

We have \(\displaystyle xy+yz+zx=4x+4y+4z\), i.e. \(\displaystyle x(4-y)+y(4-z)+z(4-x)=0\). It follows that \(\displaystyle x,y,z\) cannot all be greater than 4. WLOG, assume \(\displaystyle x\leq4\) and consider each of the cases \(\displaystyle x=1,2,3,4\) separately, thus reducing the problem to a two-variable one for each case.
 
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