USAMO 1997 Problem 5

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 $a+b+c=\sqrt{s}$, such that $0\lt a,b,c\lt\sqrt{s}$. For all $0\lt x\lt\sqrt{s}$, $f(x)$ 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 $\frac{81}{26}-4\le0\le\frac{a^3+b^3+c^3-3abc}{abc}=\frac{(a+b+c)((a-b)^2+(a-c)^2+(b-c)^2)}{2abc}$ Note by Cody Johnson
6 years, 1 month ago

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

- 6 years, 1 month ago

Answer You are absolutely right ...

- 6 years, 1 month ago

Yeah,no flaws.Great work.

- 6 years, 1 month ago

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

Staff - 6 years, 1 month ago

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

Well, if it holds for $\{a,b,c\}$ such that $a+b+c=\sqrt{s}$, 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 $\frac{\sqrt{s}}{p}$, we have $\{at,bt,ct\}=\{a',b',c'\}$ with $t=\frac{p}{\sqrt{s}}$, so it holds for all positive real $\{a,b,c\}$.

- 6 years, 1 month ago

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 $s$?

Staff - 6 years, 1 month ago

OH I SEE WHAT I DID WRONG. Because it should be a constant, not a variable, for WLOG.

- 6 years, 1 month ago

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

- 6 years, 1 month ago

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).

Staff - 6 years, 1 month ago