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Inequality proof

  1. Jul 29, 2009 #1
    1. The problem statement, all variables and given/known data
    If a, b, c > 0, prove [tex]\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \ge 3[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I'm not so sure how to do this. Usually I would try to prove that [tex]\frac{a}{b} + \frac{b}{c} + \frac{c}{a} - 3 \ge 0[/tex] but this gets me nowhere: [tex]\frac{a^2 c + b^2 a + c^2 b - 3abc}{abc}[/tex]. I can't factorise the numerator.

    I know of a similar inequality that I can prove easily using this method, which is [tex]\frac{a}{b} + \frac{b}{a} \ge 2[/tex] but the inequality above is harder for me. Please help.
  2. jcsd
  3. Jul 29, 2009 #2
    Are a, b, and c integers?
  4. Jul 29, 2009 #3
    It doesn't matter if a, b, c are integers or not. The inequality holds for real a,b,c > 0.

    Clear denominators, divide by 3, and apply the AM-GM inequality.
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