<s>[INST] Missing values of the ratio $\frac{(x+y+z)^2}{x^2+y^2+z^2}$ Let $x,y,z$ be some positive integers. Is it true that we cannot find any positive integer $n$ for which
$$
\frac{(x+y+z)^2}{x^2+y^2+z^2}=1+\frac{2}{3n}\,\,?
$$
 [/INST] Assuming the result for smaller $x+y+z$, it is inductively true in the case that any of $x,y,z$ is divisible by 3; this answer is a work in progress.
Expand and simplify to obtain:
$$\frac{xy+yz+xz}{x^2+y^2+z^2} = \frac{1}{3n}$$
or equivalently
$$3n(xy+yz+zx) = x^2+y^2+z^2$$
But the only way a sum of three squares can be a multiple of 3 is if all of them are multiples of 3 or all of them are not multiples of 3.
Former case: writing $x=3i, y=3j, z=3k$ we obtain
$$3n(ij+jk+ki) = i^2+j^2+k^2$$
We are done by inductive hypothesis: given any solution we may make a strictly smaller solution, but we have assumed for induction that no smaller solution exists. Therefore the only solution can be $x=y=z=0$, which is outside the allowable range.
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