Given $a^3+b^3+c^3=(a+b+c)^3$, show that $a^{2n+1}+b^{2n+1}+c^{2n+1}=(a+b+c)^{2n+1}$

Question

Assume that $\{a,b,c\} \subset \Bbb R$, $n\in\mathbb N$ and $a^3+b^3+c^3=(a+b+c)^3$.

Show that $a^{2n+1}+b^{2n+1}+c^{2n+1}=(a+b+c)^{2n+1}.$

This is from a list of problems used for training a team for a math olympics. I tried to use known Newton identities and other symmetric polynomial results but without success (perhaps wrong approach). Sorry if it is a duplicate. Hints and answers welcomed.

Answer 1

$$(a+b+c)^3-a^3-b^3-c^3=\sum_{cyc}(3a^2b+3a^2c+2abc)=3(a+b)(a+c)(b+c)$$ and the rest is smooth.