<s>[INST] Prove that $a$ and $b$ are rational numbers If $a+b$, $a^2+b$ and $b^2+a$ are rational numbers and $a+b\neq 1$ then $a$ and $b$ are rational.
I try, sum the expresions but I only got that $a^2+b^2$ and $ab$ are rational.
Any suggestion?
 [/INST] Write : $a^2 +  b - (b^2 + a) =(a-b)(a+b-1)$ , this is rational. Since $a+b$ is rational, so is $a+b-1$. It is non-zero, therefore $a-b$ is rational being the quotient of two rationals, the latter non-zero.
Of course, if $a+b$ and $a-b$ are rationals, so are $a$ and $b$.
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