Tricky Ineq

Tricky Inequalities is essentially a unit about inequalities that require more cleverness, yet still depend on techniques from Standard Inequalities. It is quite a fun unit, though a bit on the harder side of D (due to having to think more 🙂). One nice thing: while working on the problem set, most problems feel like you are out in the wild; the only hint you get is that the problem is an inequality (duh!). If you enjoyed this unit a lot, you can consider trying the Z level Hard Inequalities. This unit is shorter than typical, with fewer problems and a lower clubs cutoff.

Should I do this unit?

Of course. It really is fun, and it helps you practice algebraic manipulation skills along with thinking before working. Also, "Standard" inequalities don't show up that much on Olympiads nowadays because people are so comfortable with them; if you do see an inequality in contest, it will likely fall under this category now. You should probably have completed either Standard Inequalities or both of the B units Basic Inequalities and Function Inequalities before doing this unit (it assumes you know the content from those); alternatively, working through Olympiad Inequalities from Evan's website is fine.

Philosophy

You've tried AM-GM, Cauchy, Holder, and the $\frac{x}{y}\frac{y}{z}\frac{z}{x}$ substitution. You've tried Jensen, Karamata, and Tangent line trick. You have even tried (gasp!) n-1 EV and outright calculus bash. Yet the problem is STILL so elusive! Is there some elusive condition? Is there a dumb equality case that is not $x_1=x_2=...=x_n$? Well, that's what the problems in this unit are all about. The standard techniques still work, but a clever use for them needs to be found. Usually, some substitutions and algebraic manipulation of the inequalities are necessary before standard techniques finish. Other times, standard inequalities need to applied in strange, nonhomogenous manners (prove $(x^3+2)(y^3+2)(z^3+2) \geq (x+y+z)^3$ for me on positive reals please). Discovering unconventional equality cases is of paramount importance in this unit, as these will also guide you towards the correct applications of standard techniques.

Notable problems

1. 2013 ELMO 2: A great walkthrough, and a good example of how to use standard tricks in nontrivial manners to deal with even abominable conditions.
2. 2004 IMO 4: Do the modified version (P2) instead. This is a great example of dealing with strange equality cases.
3. 2007 CGMO 6: A nice example of substitutions and then standard methods.
4. 2008 ISL A5: Required on DAX. A great problem; I won't spoil it any further.
5. 2004 USAMO 6: Don't be scared by the geo. Figure out what direction the inequality is supposed to point in, and just bash it out using inequality techniques.
6. 2003 ISL A6: An amazing problem, per Evan. You should at least try this problem before submission.