Problem 12 (angle bisectors): Always keep thinking about if there is new information that can be found as a result of finding something else out.
Problem 13 (algebra): If something seems to be complicated spend some time looking for an alternative approach first before continuing with it.
Problem 20 (binary operations): When any special operation occurs, consider its relationship with common ones like +, /, -, *, etc. Sometimes they will be related.
Problem 21 (geometry): $\cos(3x)=4\cos^3(x)-3\cos(x)$. Also, arc lengths can be used to find equal angles, which can lead to similar triangles.
Problem 23 (geometry): If you are trying to use area geometry to solve a problem with 3 variables, try fixing one of them to use 2D geometry before moving on to 3D.
Problem 24 (inequalities): Any problem that involves minimums or maximums will very likely use inequalities.
Problem 25 (number theory): For big problems always make sure you understand the problem correctly (for this one I accidentally read it as cutting off all trailing digits except for one), otherwise it’s just a waste of time. For big problems, start with smaller cases and those cases will hopefully lead to generalizations.