There are many math competitions in waterloo (Pascal, Cayley, Euclid, Fryer, Galois, Hypatia, Gauss, and Fermat). In each of them, you have math questions that you need to answer. Most of the problems are not difficult but still, you need a lot of preparation to be successful. With the math you learn in school, you are not able to solve many problems and get a good result out of these exams. Here are core topics for all of these competitions:
1- Number theory: Gcd, Lcm, Euclid division algorithm, Modulo arithmetic, divisibility, Fermat’s little theorem, Euler theorem, and diophantine equations
2- Combinatorics: Permutation and combination (with repetition problem section), circle computation, binomial theorem, sets, graph theory, recursive counting.
3- Algebra: Polynomials, Quadratic equations, root coefficient relationship, maxima and minima, Solving equations, factoring.
4- Geometry: Solutions of the triangle, Ptolemy theorem, Ceva’s theorem, Menelaus theorem, and other theorems.
5- Problem-solving techniques: Induction, proof by contradiction.
For all of these competitions (Pascal, Cayley, Euclid, Fryer, Galois, Hypatia, Gauss, and Fermat), you need these topics.