<s>[INST] Find the interval of $a^2+b^2+c^2$ Given that the system of equations $x=cy+bz,y=az+cx,z=bx+ay$ has non-zero solutions and at least one of a,b,c is a proper fraction, then find the interval in which value of $a^2+b^2+c^2$ lies($a,b,c\in R$).
MY APPROACH:
I applied the cramer's rule and got the equation as $a^2+b^2+c^2=1-2abc$. Now how do i find the range of $a^2+b^2+c^2?$
 [/INST] Since one of the variables is a proper fraction, let us set, without loss of generality, $c = n/m$ where $0 < n < m$ and $n,m\in \mathbb{N}$. Then we have
$$a^2 + \dfrac{2n}{m}ab + \dfrac{n^2}{m^2}  + b^2 - 1= 0.$$
Solving for $a$ we get
$$a = \dfrac{-\dfrac{2n}{m}b \pm \sqrt{\dfrac{4n^2}{m^2}b^2 - \dfrac{4n^2}{m^2} - 4b^2 + 4}}{2} = -\dfrac{n}{m}b \pm \sqrt{(1-b^2)\left(1 - \dfrac{n^2}{m^2}\right)}.$$
Then $-1 \leq b \leq 1$. Similarly, $-1\leq a \leq 1$ which would impose a strict upper bound of $a^2 + b^2 + c^2$ of $3$.
Indeed taking $m = n+1$ and making $n$ arbitrarilly large we can consider the solution
$$a = 1,\qquad b = -\dfrac{n}{n+1}, \qquad c = \dfrac{n}{n+1}$$
and
$$a^2 + b^2 + c^2 = 1 + 2\dfrac{n^2}{(n+1)^2},$$
which can be as close to $3$ as one desires.
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