Solve the following system of equations $$\large \left \{ \begin{aligned} x^2 + y^2 &= 8\\ \sqrt[2018]x - \sqrt[2018]y = (\sqrt[2019]y - \sqrt[2019]x)&(xy + x + y + 2020)\end{aligned} \right.$$
This was a question in an exam I recently took and I was stumped. Seriously, I can't think of any approach to solve this problem.
On
I'm going to assume that $x$ and $y$ are nonnegative, or the LHS of the second equation doesn't make sense. Note that $x=y$ is a solution, which yields $x=y=2$.
Assume for contradiction there exists some other solution. Then we have $$xy + x + y + 2020 = \frac{\sqrt[2018]x-\sqrt[2018]y}{\sqrt[2019]y-\sqrt[2018]x}.$$ Note that the LHS is positive, and since $x \neq y$, the RHS is negative (since either $x>y$ in which the numerator is positive and the denominator is negative, or $x<y$ in which it's the other way around). So there can exist no solutions where $x\neq y$.
Looking at the domain of the equations, we can conclude that $x \ge 0$ and $y \ge 0$. Also, if $x \ne y$, the second equation does not have solutions as both sides will have different signs. ( If $x>y$, $\sqrt[2018]x - \sqrt[2018]y >0 $ and $\sqrt[2019]y - \sqrt[2019]x<0$ and vice versa)
Thus, $x=y=2$ is the only solution.