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...Soi for ioi syllabus fundamentals digits numbers fractions percentages etc sets and set operations inclusion exclusion complements cartesian products relations reflexivity symmetry transitivity equivalence total linear order lexicographic functions surjections injections inverses composition higher structures including vectors matrices geometry comparison arithmetic equations properties of integers basics number theory combinatorics d logic logical proof boolean values their connectives or not xor truth tables universal existential quantification notions implication converse inverse contrapositive negation contradiction direct mathematical induction strong complete problem solving strategies separation concerns structured decomposition divide conquer algorithmic design variables types expressions assignment conditional iterative control input output running an algorithm recursive mutually data structure series thinking asymptotic analysis upper complexity bounds informally if possible ...