Problem 5 from the 1997 USAMO seemed easy. Do you think my proof is correct?

Let \(s=a^3+b^3+c^3+abc\) and \(f(x)=\frac{1}{s-x^3}\). Since the terms in the inequality are homogeneous, assume without loss of generality that \(a+b+c=\sqrt[3]{s}\), such that \(0\lt a,b,c\lt\sqrt[3]{s}\). For all \(0\lt x\lt\sqrt[3]{s}\), \(f(x)\) is convex by the second derivative test. We have \(f(a)+f(b)+f(c)\le3f(\frac{a+b+c}{3})=\frac{81}{26s}\) by Jensen's Inequality. To prove that \(\frac{81}{26(a^3+b^3+c^3+abc)}\le\frac{1}{abc}\), we have \(\frac{81}{26}\le\frac{a^3+b^3+c^3+abc}{abc}\) and \[\frac{81}{26}-4\le0\le\frac{a^3+b^3+c^3-3abc}{abc}=\frac{(a+b+c)((a-b)^2+(a-c)^2+(b-c)^2)}{2abc}\]

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## Comments

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TopNewestGod I wish I knew how to do problems like this...

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Answer You are absolutely right ...

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Yeah,no flaws.Great work.

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Can you explain the WLOG? In particular, if \(a=b=c\), what would the WLOG values be?

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Do you need to explain your choice of WLOG on these exams?

Well, if it holds for \(\{a,b,c\}\) such that \(a+b+c=\sqrt[3]{s}\), we can show it holds for \(\{at,bt,ct\}\) where \(t\) is a positive real:

\[\sum_{cyc}\frac{1}{(at)^3+(bt)^3+abct^3}\le\frac{1}{abct^3}\]

Multiplying by \(t^3\) gives the desired result. So, for a set \(\{a',b',c'\}\), let \(p=a'+b'+c'\). If we multiply both sides by \(\frac{\sqrt[3]{s}}{p}\), we have \(\{at,bt,ct\}=\{a',b',c'\}\) with \(t=\frac{p}{\sqrt[3]{s}}\), so it holds for all positive real \(\{a,b,c\}\).

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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 \(s\)?

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It's a valid proof, but I would argue that the second derivative test isn't necessarily fair game on the USAMO.

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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).

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