IndoMathXdZ wrote: Does the system of equation \begin{align*} \begin{cases} x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\ x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\ x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3 \end{cases} \end{align*}admit a solution in integers such that the absolute value of each of these integers is greater than $2020$? Yes. Choose for example : $(x_1,x_2,y_1,y_2,y_3,y_4)=(5x,-5x,3x,-3x,4x,-4x)$ for $x$ great enough

I want to motive this a little further: note that if a (coordinatewise) non-zero solution is found, we are done: if $(x_1,\dots,y_4)$ works, so is $(Nx_1,\dots,Ny_4)$, for any $N$ integer. Now, $x_2=-x_1$, $y_2=-y_1$ and $y_4=-y_3$, one can ensures the first and third conditions hold trivially. For the second, it boils down ensuring $x_1^2=y_1^2+y_3^2$, but for this one can simply choose a Pythagorean triple.