Problem 5 from the 1997 USAMO seemed easy. Do you think my proof is correct?
Let s=a3+b3+c3+abc and f(x)=s−x31. Since the terms in the inequality are homogeneous, assume without loss of generality that a+b+c=3s, such that 0<a,b,c<3s. For all 0<x<3s, f(x) is convex by the second derivative test. We have f(a)+f(b)+f(c)≤3f(3a+b+c)=26s81 by Jensen's Inequality. To prove that 26(a3+b3+c3+abc)81≤abc1, we have 2681≤abca3+b3+c3+abc and 2681−4≤0≤abca3+b3+c3−3abc=2abc(a+b+c)((a−b)2+(a−c)2+(b−c)2)
This discussion board is a place to discuss our Daily Challenges and the math and science
related to those challenges. Explanations are more than just a solution — they should
explain the steps and thinking strategies that you used to obtain the solution. Comments
should further the discussion of math and science.
When posting on Brilliant:
Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.
Markdown
Appears as
*italics* or _italics_
italics
**bold** or __bold__
bold
- bulleted - list
bulleted
list
1. numbered 2. list
numbered
list
Note: you must add a full line of space before and after lists for them to show up correctly
Do you need to explain your choice of WLOG on these exams?
Well, if it holds for {a,b,c} such that a+b+c=3s, we can show it holds for {at,bt,ct} where t is a positive real:
cyc∑(at)3+(bt)3+abct31≤abct31
Multiplying by t3 gives the desired result. So, for a set {a′,b′,c′}, let p=a′+b′+c′. If we multiply both sides by p3s, we have {at,bt,ct}={a′,b′,c′} with t=3sp, so it holds for all positive real {a,b,c}.
@Cody Johnson
–
Yes, you have to explain the WLOG, and make sure that it indeed is WLOG. Your current application is not standard or typical, and you should review how to apply it properly. The reason I'm pointing it out, is that WLOG does not hold, but you have instead restricted the possible sets.
Note that you defined the variable s twice, and need both scaled equations to hold. You have not answered how to deal with the case that a=b=c=1. What is the scaling value of t and the corresponding value of s?
I am pointing out that this is not a valid proof. Review the WLOG argument, in which he actually introduces a restrictive condition.
The second derivative test (for convexity) is fair game on the USAMO. However, make sure that you write out the full statements, instead of just claiming that it works (as was done above).
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
*italics*
or_italics_
**bold**
or__bold__
paragraph 1
paragraph 2
[example link](https://brilliant.org)
> This is a quote
\(
...\)
or\[
...\]
to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
God I wish I knew how to do problems like this...
Log in to reply
Answer You are absolutely right ...
Yeah,no flaws.Great work.
Log in to reply
Can you explain the WLOG? In particular, if a=b=c, what would the WLOG values be?
Log in to reply
Do you need to explain your choice of WLOG on these exams?
Well, if it holds for {a,b,c} such that a+b+c=3s, we can show it holds for {at,bt,ct} where t is a positive real:
cyc∑(at)3+(bt)3+abct31≤abct31
Multiplying by t3 gives the desired result. So, for a set {a′,b′,c′}, let p=a′+b′+c′. If we multiply both sides by p3s, we have {at,bt,ct}={a′,b′,c′} with t=3sp, so it holds for all positive real {a,b,c}.
Log in to reply
Note that you defined the variable s twice, and need both scaled equations to hold. You have not answered how to deal with the case that a=b=c=1. What is the scaling value of t and the corresponding value of s?
Log in to reply
It's a valid proof, but I would argue that the second derivative test isn't necessarily fair game on the USAMO.
Log in to reply
I am pointing out that this is not a valid proof. Review the WLOG argument, in which he actually introduces a restrictive condition.
The second derivative test (for convexity) is fair game on the USAMO. However, make sure that you write out the full statements, instead of just claiming that it works (as was done above).