(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold? (b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold?
Source: Dutch IMO TST 2018 day 2 p1
Tags: system of equations, algebra
(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold? (b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold?