MathNet 2024-part-01 — 168 problems (EN–VI) Question 143

Do there exist mutually distinct real numbers $a$, $b$, $c$ such that the numbers $a^2 + b$, $b^2 + c$, $c^2 + a$ are equal, in some order, to the numbers $a + b^2$, $b + c^2$, $c + a^2$?