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How exactly do I solve this problem? (Source: 1984 British Math Olympiad #3 part II)

\begin{equation*} \bigl(\frac{a}{b}\bigr)^4 + \bigl(\frac{b}{c}\bigr)^4 + \bigl(\frac{c}{d}\bigr)^4 + \bigl(\frac{d}{e}\bigr)^4 + \bigl(\frac{e}{a}\bigr)^4 \ge \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{e}{d} + \frac{a}{e} \end{equation*}

There's not really a clear-cut way to use AM-GM on this problem. I've been thinking of maybe using the Power Mean Inequality, but I don't exactly see a way to do that. Maybe we could use harmonic mean for the RHS?

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    $\begingroup$ someone please explain why this is closed. I think I have adequately explained some strategies that I've tried. I believe I've provided enough context. $\endgroup$
    – crxyz
    Commented Dec 10, 2022 at 19:54
  • $\begingroup$ I'm kinda new around here, but I was also surprised to see it closed. Also I found the accepted solution to be very nice. $\endgroup$
    – 3rdMoment
    Commented Dec 10, 2022 at 21:48

2 Answers 2

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Applying the AM-GM $$LHS - \bigl(\frac{e}{a}\bigr)^4 = \bigl(\frac{a}{b}\bigr)^4 + \bigl(\frac{b}{c}\bigr)^4 + \bigl(\frac{c}{d}\bigr)^4 + \bigl(\frac{d}{e}\bigr)^4 \ge4 \cdot \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot\frac{d}{e} = 4\cdot\frac{a}{e} $$ Do the same thing for these 4 others terms, and make the sum $$5 LHS - LHS \ge 4 RHS$$ $$\Longleftrightarrow LHS \ge RHS$$ The equality occurs when $a=b=c=d=e$

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  • $\begingroup$ I think in the end you should have $5LHS-LHS\geq 4RHS$, since you repeat the procedure 5 times, not 4. Then everything works. :) $\endgroup$ Commented Dec 5, 2022 at 9:33
  • $\begingroup$ @Freshman'sDream You're right, I just corrected this typo. Thanks! $\endgroup$
    – NN2
    Commented Dec 5, 2022 at 9:34
  • $\begingroup$ ohhhhh ok thanks! $\endgroup$
    – crxyz
    Commented Dec 9, 2022 at 23:08
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NN2 gave a simple and very elegamt proof. I tried another way.

What is the minumum of the function $f(x_1,x_2,x_3,x_4,x_5)=\sum_{i=1}^{5}(x_i^4-x_i^{-1})$ with domain $\Bbb{R}^{5+}$, subject to the constraint equation $x_1x_2x_3x_4x_5=1$?

The system of a Lagrange multplier $\lambda$ gives the equations $4x_i^3+x_i^{-2}=\lambda x_i^{-1}$ for all $i=1,2,3,4,5$. From these equations we have $$4x_ix_j(x_i^4-x_j^4)=x_i-x_j$$ for all $i,j.$ I am stuck. Any ideas?

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