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Let us consider three sequences $(a_n)_{n\ge1}$, $(b_n)_{n\ge1}$ and $(c_n)_{n\ge1}$ having the properties:

  • $a_{n},\ b_{n},\ c_{n}\in\left(0,\ \infty\right)$
  • $a_{n}+b_{n}+c_{n}\ge\frac{a_{n}}{b_{n}}+\frac{b_{n}}{c_{n}}+\frac{c_{n}}{a_{n}}\ \forall n\ge1$
  • $\lim\limits_{n\to\infty}a_{n}b_{n}c_{n}=1 $

Prove that $$\lim_{n\to\infty}\frac{a_{n}+b_{n}+c_{n}}{a_{n}b_{n}+b_{n}c_{n}+c_{n}a_{n}}=1 $$

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    I first guessed that $a_n,b_n,c_n\to1$, but now I think it might be wrong. I can only show that, when $abc=1$, we have $a/b+b/c+c/a\ge a+b+c$. It suffics to prove that $a/b+a/b+b/c\ge3a$, where AM-GM works.2012-06-24
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    What is the purpose of the word "own" in the title?2012-06-25
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    @ByronSchmuland Maybe he's finding a pretty proof for his own result.2012-06-26
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    @ByronSchmuland: I think this problem may be original. I sent this problem to a elementary math magazine from Romania.2012-06-26
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    @FrankScience: I think I have a nice proof for the problem.2012-06-26
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    @ClaudiuMindrila As you can see, in our community, the information that your problem is original is not posted by a code word in the title. That information, and the fact that you have a proof and are looking for alternatives goes into the body of the question.2012-06-26
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    @ClaudiuMindrila I should add: welcome to MSE! You have nice problems.2012-06-26

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