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I really like inequality problems, but I've found that I have severe difficulties with ones that should be simple. I can prove something simple like this

For positive $a, b, c$ with $a + b + c = 6$ show that $$\left(a + \frac{1}{b} \right)^2 + \left(b + \frac{1}{c} \right)^2 + \left(c + \frac{1}{a} \right)^2 \geq \frac{75}{4} $$

but I still find problems like

Let $a, b, c$ be positive reals. Prove that $$ \frac{a^2}{\sqrt{4a^2 + ab + 4b^2}} + \frac{b^2}{\sqrt{4b^2 + bc + 4c^2}} + \frac{c^2}{\sqrt{4c^2 + ca + 4a^2}} \geq \frac{a + b + c}{3}$$

intractable. To give you an idea of my competence, I've been able to barely solve the following problem:

Show that for positive $a, b, c$, we have $$ \frac{a^2}{(a + b)(a + c)} + \frac{b^2}{(b + c)(b + a)} + \frac{c^2}{(c + a)(c + b)} \geq \frac{3}{4}$$

My solution involved multiplying out the denominators, playing around with an obscure algebraic identity, and then finally getting a bound with AM-GM, which is not what I believe the writer had in mind.

My question is, what sort of problems should I attempt, and how can one improve at inequality problems in general?

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    $\begingroup$ I think if one have learned calculus, we can regard it as a multi-variate function. Some optimization methods can be helpful, say, partial derivative and Lagrange's multiplier method,etc, to find the optimized point and value of the function. $\endgroup$ – Robert Fan Mar 15 '14 at 14:13
  • $\begingroup$ This is not really what I have in mind, sadly. While the method of Lagrange multipliers can be useful in inequality problems, it is hardly ever elegant or convenient and generally requires immense computational fortitude. $\endgroup$ – Ayesha Mar 15 '14 at 14:31
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Searching for "inqualities" on Google gives lots of hits. The Wikipedia article has lots of references. And Lohwater's Introduction to inequalities is very thorough on the type of inequalities you describe (don't be fooled by the huge PDF, it is a short typewritten manuscript).

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