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# Inequality

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1. How can I show that for all positive integers a, b, and c, a/b + b/c + c/a >= 3?
2. (Original post by HapaxOromenon3)
How can I show that for all positive integers a, b, and c, a/b + b/c + c/a >= 3?
AM-GM inequality case n=3.

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3. (Original post by physicsmaths)
AM-GM inequality case n=3.

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Yes, that seems to work, with the list of integers being (ab^2, bc^2, ca^2). Thanks.
4. (Original post by HapaxOromenon3)
Yes, that seems to work, with the list of integers being (ab^2, bc^2, ca^2). Thanks.
You can do it directly.
Multiplying these terms gives 1 so LHS/3>=1 hence he inequality.
Alternatively note that a->ka b->kb etc the inequality is not changed so homegenous, so you may assume something which gives a different type of solution albeit longer but more satisfying.

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5. The homogenous stuff is just food for thought for other inequalities where such a fact is very useful and used to the absolute extreme.

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