One problem solving approach which I adopted around the time that I was very frequently reading Terry's blog was to personify the problem as a combatant. For example, when trying to prove an inequality that seemed straightforward I would focus on how it could be false, and to then consider what structure the resulting 'conspiracy' of variables would have to have. I have a feeling that this is something that Terry did (and I emulated), but I can't remember a specific post.

He also has a very nice post on 'amplification and arbitrage' as tricks for strengthening inequalities.