I've lately been noticing a trend with the skills involved for various topics in math. I'm basing my thoughts solely my own observations, but here're my two cents. Roughly, most olympiad problems fall into a spectrum from spontaneous to un-spontaneous.

Roughly, combo, algebra, NT, and geo fall onto the spectrum from spontaneous (left) to un-spontaneous (right) in this respective order.

Un-spontaneous problems are the kind where knowledge of theory/configurations can be especially useful. This is especially true in geometry. Complex configurations being based on the most basic of which can be proved with simple tools like angle chasing + cyclic quads + radical axis, etc. But often times, recognition of a certain configuration and its properties automatically gives you a step up above others who've not seen it, making it easier for you to derive further properties and ultimately, solve the problem. This is also the main benefit of getting Lemmas in Oly Geo and why classes like Geo 3 are incredibly useful.

Spontaneous problems are problems where intuition and heuristics come into play. These problems demand more ingenuity and problem-solving skills than those on the right end, which require more conceptual/theoretical understanding. These problems are mostly combinatorial, and a particular characteristic of the solutions to these kinds of problems is that they're almost always easily understandable, but hard to motivate. These are also the problems I like to use as examples when I'm explaining what kind of math I'm doing to my less-mathematically-inclined family members / acquaintances (since the problem statements are easier to understand and think about).

There're few to none theorems/facts to know that will give you a step up other than general heuristics to follow (e.g. pigeonhole, looking at max/min element, finding invariants). Useful strategies include experimentation, wishful thinking, working backwards (finding a penultimate step), and others that are much easier said than done... but nonetheless the ideas can be incredibly simple and beautiful.

Number theory and algebra are in between, though I would argue algebra is more spontaneous than number theory (where there're many techniques involving mods, congruences, exponentials, and high power tools like LTE, V.J.), and also because the more classical inequalities are being replaced by nontraditional ones and the more spontaneous FE's.

This spectrum is also why you often hear people saying "combo is my worst subject" or (on the opposite) "geo is my worst subject," but less frequently you hear people say that for NT or Alg. It would be interesting to analyze the non-mathematical differences between these (roughly categorized) lefties and righties.

Personally, I'm inclined towards the right. Combo problems and those aha! ideas are honestly more fun and interesting, but for some reason my first response to most problems is to try to relate it to some theoretical knowledge / similar results that I've seen before, an example being that nowadays my first instinct is to try projective as opposed to basic tools on geo problems (which I've gotten rusty on oops). This is somewhat ironic considering that creativity is one of my top strengths.

This, I realized, might be more of a consequence of rigid and algorithmic nature of school, the aptitude for which requires a fundamental different part of the brain than that of those naughty boys and girls who're the most creative problem solvers (esp. lefties).

Just last week, I was going through a shortcut to the library (which I discovered two months ago that saves a minute of my time every time I go to the library), when suddenly the vice principal halted me and acerbically said "This is not the entrance to the library. Please go around and enter through the main entrance where everyone else comes in." Right as I was about to voice out my protest, I decided it not worth the time to fall into her (annoyingly dangerous) radar and give her a rebellious stare instead, before turning around. Rules are meant to be bent. ...and even if school teaches you the opposite, this is the mindset I (and everyone else) should have when solving problems.