Inequality with condition $xy+yz+zx=3xyz$

For discussing Olympiad Level Algebra (and Inequality) problems
rah4927
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Inequality with condition $xy+yz+zx=3xyz$

Unread post by rah4927 » Thu Aug 04, 2016 2:08 pm

Let $x,y,z$ be positive real numbers such that $xy+yz+zx=3xyz$ . Prove that :

$$x^2y+y^2z+z^2x \ge 2(x+y+z)-3$$

rah4927
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Re: Inequality with condition $xy+yz+zx=3xyz$

Unread post by rah4927 » Fri Aug 05, 2016 11:12 am

The thing that always bums me out in ineq probs is the weird constant terms (often +1,-1,+3,-3) . in this prob, its easy to get rid of that; $3 = \frac1x + \frac1y + \frac1z $ . Of course, minus signs never help, so we move these to the left.

At this point, you will probably be bothered by the annoying coefficient of $(x+y+z)$. I mean, $2$ is such an annoying thing to have in a three variable inequality. So we add $x+y+z$ to both sides . Now our inequality looks like this.

$$x^2y+y^2z+z^2x+\frac1x + \frac1y + \frac1z +x+y+z \ge 3(x+y+z)$$ .

There are $9$ terms on the left, and a coefficient of $3$ on the right. This indicates that we should apply AM-GM in groups of three. By picking $x^2y+x+1/y\ge 3x$ cyclically and summing them up, we end up with the desired inequality.

Mehedi Hasan Nowshad
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Re: Inequality with condition $xy+yz+zx=3xyz$

Unread post by Mehedi Hasan Nowshad » Wed Nov 02, 2016 12:17 am

I think 2 is not annoying here as you can simply have by AM-GM,
\[ \sum_{cyc} (x^2y+\dfrac{1}{y}) \geq 2\sum_{cyc} x \]
the desired inequality!!!
"Failure is simply the opportunity to begin again, this time more intelligently."
- Henry Ford

Levid0
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Re: Inequality with condition $xy+yz+zx=3xyz$

Unread post by Levid0 » Fri May 28, 2021 3:41 am

Hi, I’m new here but im trying this exact question from 2017 PAMO!

I know the thread is years old, but can anyone help me understand this am—gm and how these statements are true? I get the typical am-gm like( x1+x2+x3+…+xn/n)>=nth root(x1x2x3…xn), but x²y+1/y? Looks completely different. I’m very new to this stuff so if you say “read this book” thats fine :D

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Anindya Biswas
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Re: Inequality with condition $xy+yz+zx=3xyz$

Unread post by Anindya Biswas » Mon May 31, 2021 11:26 pm

Levid0 wrote:
Fri May 28, 2021 3:41 am
Hi, I’m new here but im trying this exact question from 2017 PAMO!

I know the thread is years old, but can anyone help me understand this am—gm and how these statements are true? I get the typical am-gm like( x1+x2+x3+…+xn/n)>=nth root(x1x2x3…xn), but x²y+1/y? Looks completely different. I’m very new to this stuff so if you say “read this book” thats fine :D
\[\dfrac{x^2y+\frac{1}{y}}{2}\geq \sqrt{x^2y\cdot\frac{1}{y}}\]
\[\Rightarrow x^2y+\frac{1}{y}\geq2x\]
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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Anindya Biswas
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Re: Inequality with condition $xy+yz+zx=3xyz$

Unread post by Anindya Biswas » Tue Jun 01, 2021 12:23 am

Try reading this amazing note :
Applying AM_GM like a boss.pdf
(197.75KiB)Downloaded 176 times
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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