Problem

Source: Polish MO Finals P4 2024

Tags: algebra, system of equations, parameterization, algebra proposed



Do there exist real numbers $a,b,c$ such that the system of equations \begin{align*} x+y+z&=a\\ x^2+y^2+z^2&=b\\ x^4+y^4+z^4&=c \end{align*}has infinitely many real solutions $(x,y,z)$?