a_507_bc wrote: Denote a set of equations in the real numbers with variables $x_1, x_2, x_3 \in \mathbb{R}$ Flensburgian if there exists an $i \in \{1, 2, 3\}$ such that every solution of the set of equations where all the variables are pairwise different, satisfies $x_i>x_j$ for all $j \neq i$. Find all positive integers $n \geq 2$, such that the following set of two equations $a^n+b=a$ and $c^{n+1}+b^2=ab$ in three real variables $a,b,c$ is Flensburgian. We have $b=a-a^n$ and $c^{n+1}=ab-b^2=a^{n+1}-a^{2n}$. For even $n$ we have $b\leq a$ and $c^{n+1}\leq a^{n+1}$ since squares are non-negative. Since $n+1$ is odd, we have $c\leq a$. So $a=\max\{a,b,c\}$ for every solution for odd $n$. So the system is Flensburgian for even $n$. For odd $n$ we first choose $a=1-\varepsilon$ and the negative solution for $c$. (We can do this since $(1-\varepsilon)^{n+1}>(1-\varepsilon)^{2n}$.) Then we have $c<b<a$. But if we choose $a=-\varepsilon$ and the positive solution for $c$ (we can do this since $\varepsilon^{n+1}>\varepsilon^{2n}>0$) we get $c>b>a$. So the system is not Flensburgian for odd $n$.