# How to get better at inequality problems?

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?

• 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. – Robert Fan Mar 15 '14 at 14:13
• 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. – Ayesha Mar 15 '14 at 14:31