Alg Manip
Problems that purely involve algebraic manipulations without some other context, e.g. solving rather arbitrary equations or systems of equations.
Algebraic Manipulation is a general mix of algebra problems that do not involve evoking any specialized tools (although AM-GM can be useful, but don't expect say Jensen's or Schur's). Trig subs, equalities that are actually inequalities, and plain manipulation are techniques mentioned in the walkthrough portion.
Philosophy
The general glue that holds the problems together as a group is that they are algebra problems without any high-tech tools and are not functional equations. That means that the unit is very adhoc and not entirely cohesive across the problem set.
Across the problem set, expect a decent amount of "pure" inequalities, ones that are mostly polynomial-ish and low-power. But most of it is just "plain" algebra.
As for advice, for some problems it may be best to forget that they're under an algebra unit. Also, when solving systems of equations, if there should be the existence of difficult complex solutions that probably means it's an equality hidden as an inequality, as then it's possible to apply the order structure of the reals.
Notable Problems
Because of the loose nature of this problem set, there are many subgroups one can try to construct. Here are some problems representative of the more prominent subgroups.
- Shortlist 2006 A2: Expect to analyze many strange sequences.
- IMO 2018/2: A polynomial-like equality that is actually an inequality.
- Shortlist 2018 N3: Some problems involve a free-ish construction in some way.