I sympathize with the question, but I fear it is too broad, because if you make a search for "matrix" or "integral" on Art of Problem Solving (the olympiad section) you will get lots and lots of replies. Whole fields of olympiad mathematics are basically taken from advanced mathematics. Zero-sum combinatorics, coding theory, inequalities, more inequalities (with both an analysis and a linear algebra proof), even some functional equations are prone to advanced approaches (not advanced as in requiring 5 years of reading EGA, but advanced as in not taught in school - which is a rather unnatural divide to take anyway). A good way to find such problems is to look out for Russian mathematical contest problems (particularly the olympiad of the St. Petersburg Lyceum 239, if you can get your hands on their problems), since these are very often authored by academics rather than by teachers, educators or full-time olympiad trainers.