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$a;b;c;x;y;z \in \mathbb{R}$ such that : \begin{matrix} a+b+c=x+y+z & \\ a^2+b^2+c^2=x^2+y^2+z^2 & \\ a^3+b^3+c^3=x^3+y^3+z^3 & \end{matrix}

Prove that : $a^n+b^n+c^n=x^n+y^n+z^n$; $\forall n\in \mathbb{N}$

P/s : I don't have any ideas about this problem..!!

Thanks :)

Asimov
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Lê Tấn Khang
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  • Hmm. Note that $3abc=a^3+b^3+c^3-\frac12(a+b+c)^3+\frac12(a+b+c)(a^2+b^2+c^2)=3xyz$, which should say something about the signs. As soon as we know all variables have the same sign, it follows from Hölder's inequality/Cauchy-Schwarz. I don't have a complete solution yet. – Bart Michels Oct 25 '14 at 15:54
  • Can you show me how to do next !? I don't understand !! :( – Lê Tấn Khang Oct 25 '14 at 15:57
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    It should have been $3abc=a^3+b^3+c^3+\frac12(a+b+c)^3-\frac32(a+b+c)(a^2+b^2+c^2)$. But more is true, see Macavity's answer. – Bart Michels Oct 25 '14 at 16:00

1 Answers1

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Hint:

Let $p_1=a+b+c, \; q_1 = ab+bc + ca,\; r_1 = abc, \; p_2 = x+y+z, \; q_2 = xy+yz+zx, \; r_2 = xyz$. Then it is easy to show that: $$p_1 = p_2, \quad q_1 = q_2, \quad r_1 = r_2$$ Further, $a^n+b^n + c^n$ and $x^n + y^n + z^n$ are both symmetric functions and hence can be expressed in terms of $p_1, q_1, r_1$ and $p_2, q_2, r_2$ respectively in exactly the same manner... so they are equal.

Macavity
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  • I seem to lack the algebraic background to get this argument. Is this not just about the polynomial $f(u,v,w) = u^n + v^n + w^n$ applied to two triplets of numbers $(a,b,c)$ and $(x,y,z)$? Yes $f$ is symmetric but you mention two potentially different symmetric functions, I fail to recognise them. – mvw Oct 25 '14 at 16:25
  • @mvw you're right in that the two are really the same function, and perhaps is better phrased so rather than saying they express in terms of $p_i, q_i, r_i$ in the same manner. Alternately one could make the argument that it is the same symmetric function evaluated at two triplets which are permutations of each other, being roots of the same cubic. Just different ways of showing the same thing, I suppose. – Macavity Oct 25 '14 at 16:43