<s>[INST] Find all triples of non-negative real numbers $(a,b,c)$ Find all triples of non-negative real numbers $(a,b,c)$ such that:
$a^2+ab=c$
$b^2+bc=a$
$c^2+ca=b$.
This question was problem number of 3 in the RMO(India) Olympiad in 2019 held on $10^{th}$ November. link  .

My attempt-
Assume $a \geq b\geq c$,
$\therefore a^2 \geq b^2$ and $ab\geq bc$.
Adding these two, $a^2+ab\geq b^2+bc$
implies $c \geq a$.
Which can be possible only if  $a=c$, which also implies $a=b=c$.
Substituting in the base equations,
$a^2+a^2=a$
$\therefore (a,b,c)=(0,0,0)$ or $(0.5,0.5,0.5)$
I want to know if my method is correct because it seems a lot different than the one provided in the solutions.
 [/INST] Eliminating $b,c$ from our system we get for $a$ the following equation:
$$2 a^8+a^7+a^6+3 a^5+6 a^4+2 a^3-a^2-a=0$$
and this is $$a (2 a-1) \left(a^6+a^5+a^4+2 a^3+4 a^2+3 a+1\right)=0$$
</s>