<s>[INST] $a+b=c+d$ , $a^3+b^3=c^3+d^3$ , prove that $a^{2009}+b^{2009}=c^{2009}+d^{2009}$ 
Let $a, b, c, d$ be four numbers such that $a + b = c + d$ and $a^3 + b^3 = c^3 + d^3$. Prove that $a^{2009}+b^{2009}=c^{2009}+d^{2009}$.

I've got $a^2+b^2-ab=c^2+d^2-cd$.
I tried squaring or cubing it repeatedly but I didn't get what I wanted. 
Now how do I proceed? 
 [/INST] If $a=-b$ so $c=-d$ and we are done.
Let $a+b\neq0.$
Thus, $$(a+b)^3-3ab(a+b)=(c+d)^3-3cd(c+d)$$ or
$$ab(a+b)=cd(c+d)$$ or $$ab=cd$$ and the rest is smooth.
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