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 October 18th, 2018, 12:38 PM #1 Senior Member   Joined: Oct 2016 From: Arizona Posts: 193 Thanks: 34 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. Inequality tips Hi, so I'm working on an inequality problem, and the problem gives you that $a,b,c$ are real numbers and $a+b+c=3$. Using this and Cauchy Schwarz, I was able to get that $a^2+b^2+c^2 \ge 3$ and $-3 \le ab+bc+ca \le 3$. This helps out, but there is still $abc$ I would like to know something about, but can't seem to figure out any bounds on it. Does anyone have any ideas? Specifically there is a term $+18abc$ which I'd like to be $\ge 0$ and I've written that as $2(a+b+c)^2(abc)$ but we obviously can't conclude that $abc$ is \ge0 since they are real numbers. Sorry if this is confusing. I'm mainly looking for ideas on what I could do. October 18th, 2018, 01:25 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 578 Thanks: 345 Math Focus: Dynamical systems, analytic function theory, numerics It seems you are leaving out some information. You can't conclude anything about the sign of $abc$ for exactly the reasons you have stated. October 18th, 2018, 01:39 PM   #3
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Joined: Oct 2016
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Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.
Quote:
 Originally Posted by SDK It seems you are leaving out some information. You can't conclude anything about the sign of $abc$ for exactly the reasons you have stated.
I can just figure it out, thanks! October 18th, 2018, 03:14 PM #4 Member   Joined: Oct 2018 From: Netherlands Posts: 39 Thanks: 3 There are no bounds to $abc$. For instance, with a large positive $bound$ take the following values: $a = +bound$ $b = -bound+4$ $c = -1$ The special case is where $a, b, c$ are all positive real numbers. Not hard to see that $0 <= abc <= 1$ Last edited by Arisktotle; October 18th, 2018 at 03:17 PM. October 18th, 2018, 03:23 PM   #5
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Joined: Oct 2016
From: Arizona

Posts: 193
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Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.
Quote:
 Originally Posted by Arisktotle There are no bounds to $abc$. For instance, with a large positive $bound$ take the following values: $a = +bound$ $b = -bound+4$ $c = -1$ The special case is where $a, b, c$ are all positive real numbers. Not hard to see that $0 <= abc <= 1$
Okay, thank you! I will try to incorporate this term into the rest of the inequality. I appreciate it! October 19th, 2018, 10:40 AM #6 Senior Member   Joined: Oct 2016 From: Arizona Posts: 193 Thanks: 34 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. This is my 5th inequality proof and it's crazy how many tricks of the trade I learned, I had to completely change my approach! Lesson: if an approach isn't working, don't keep trying it  Tags inequality, tips Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post StillAlive Calculus 5 September 2nd, 2016 11:45 PM WWRtelescoping Advanced Statistics 4 March 13th, 2016 03:34 PM Drake Algebra 10 April 9th, 2013 02:14 PM RussellBear New Users 16 March 12th, 2012 03:34 PM

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