I'm sharing my knowledge. This is what I think you need in competitive programming.

Number Theory: You'll need modular inverse(which is derived from Fermat's little theorem). Some interesting theorems like Wilson's Theorem and CRT(Chinese remainder theorem) might be useful.

Geometry: Convex hull, Pick's theorem, calculating area of polygon, dot and cross product(and its use for CW/CCW)

Combinatorics: well... combination, permutation, repetition, PIE(principle of inclusion & exclusion), using generating functions to count things

Polynomials: FFT(you don't have to understand the full thing about signal decomposition)

It looks like some math olympiad syllabus, but those things might help you if you're not familiar with them. Also needed is your instinct, to apply algorithms and theorems to efficiently solve the problems ;)