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Let $x, y, z$ be real numbers such that $-1< x + y + z < 1$ and $x^2 + y^2 + z^2 < 1$.

Prove the inequality or give a counter example:

$$(x^2 + 2yz)^2 + (y^2 + 2xz)^2 + (z^2 + 2xy)^2 < 1$$

I do not know if it is true or not.

Jyrki Lahtonen
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Vlad
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    I removed the real-analysis tag because that has more to do with analytical/topological and more advanced concepts of the system of real numbers. IMO inequality is a sufficient tag. Inequalities are a recurring theme in math competitions, and we have members well versed in those techniques. It is possible that methods of (multi-variable) calculus come to the fore. Should that be the case here an appropriate tag can be added. – Jyrki Lahtonen Dec 24 '14 at 09:17
  • Maybe it's $\dots<3$? – Giulio Bresciani Dec 24 '14 at 10:16
  • @barakmanos Your counter example does not work ($x^2+y^2+z^2=1.62$). If you go just below $1/\sqrt{2}$ for $x$ it works (see my answer below). – mickep Dec 24 '14 at 11:10

1 Answers1

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I dont think this is true. Take $x=0$, $y=1/\sqrt{2}-\epsilon$ and $z=-1/\sqrt{2}+\epsilon$, where $\epsilon$ is small.

It might be easier to see if you just let $x=0$ and $y=-z$. Then the first inequality is automatically satisfied. The second inequality becomes $2z^2<1$. The third one $6z^4<1$. It is clear that we can find $z$ satisfying the second but not the third.

mickep
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  • I don't think you are right. 6z^4<1 has solutions. obviously z=0 satisfies all inequalities. x=y=z=0 satisfies all three initial inequalities. But the questions is - is this true for all real number given than −1<x+y+z<1 and x2+y2+z2<1 – Svetlin Mladenov Dec 24 '14 at 10:19
  • Please read more carefully what I write. I say that we can find a $z$, not that it is false for all $z$. – mickep Dec 24 '14 at 10:27
  • Thank you very much. Your counter example is absolutely correct. – Vlad Dec 24 '14 at 16:10
  • I suggest that you ask this as a new question (if that is the way to do it) or edit your question (if that is the way...). I'm new to SE so I don't know... – mickep Dec 24 '14 at 19:03