INMO 2007 Problem 2

Let $n$ be a natural number such that $n=a^{2}+b^{2}+c^{2}$, for some natural numbers $a, b, c$. Prove that

$$9 n=\left(p_{1} a+q_{1} b+r_{1} c\right)^{2}+\left(p_{2} a+q_{2} b+r_{2} c\right)^{2}+\left(p_{3} a+q_{3} b+r_{3} c\right)^{2}$$

where $p_{j}$ 's, $q_{j}$ 's, $r_{j}$ 's are all nonzero integers. Further, if 3 does not divide at least one of $a, b, c$, prove that $9 n$ can be expressed in the form $x^{2}+y^{2}+z^{2}$, where $x, y, z$ are natural numbers none of which is divisible by 3 .