Students will learn to prove properties and results, and solve non-routine problems involving:
(a) Functions: e.g. graphs, symmetries, derivatives, integrals, differential equations, limiting behaviours,
(b) Sequences and series: e.g. general terms, sum, limiting behaviours, bounds.
(c) Inequalities: AM–GM inequality, Cauchy-Schwarz inequality, triangle inequality.
(d) Numbers: primes, coprimes, divisibility, greatest common divisor, division algorithm, congruences and modular arithmetic.
(e) Counting: distribution problems, Stirling numbers of the second kind, recurrence equations, bijection principle, principle of inclusion and exclusion.
(f) Mathematical language: Terms such as ‘Definition’ and ‘Theorem’, Conditional statements (such as ‘if P then Q’ and ‘P if and only
if Q’), Necessary and sufficient conditions, Existential and universal quantifiers (such as ‘there exists’, ‘for each’, ‘for all’), Logical connectives (such as ‘and’, ‘or’, ‘not’, ‘implies’), Converse, inverse, contrapositive and negation of statements, Set notation and language
(g) Methods of Proof: Direct proof, Proof by mathematical induction, Disproof by counterexample, Proof by contradiction, Proof of existence, Proof by construction, Pigeonhole principle, Symmetry principle
(h) Problem solving heuristics: Working backwards, Uncovering pattern and structure, Solving a simpler/similar problem, Considering cases, Restating the problem (e.g. contrapositive)
The above define the expected scope of content knowledge that may be assessed.
Notwithstanding the content areas defined above, students will also prove results and solve problems outside these defined areas or at the intersection of two or more such areas using their ability to understand and apply given definitions or results.
Notwithstanding the presentation of the topics in the syllabus document, it is envisaged that some examination questions may integrate ideas from more than one topic, and that topics may be tested in the contexts of problem solving and application of mathematics.

Mathematical Results
Students may use the following theorems and results without proof. In addition, they may be required to use the ideas in the proofs of these theorems and results to solve other problems. This means that questions should not be asked to prove these results directly.
(i) AM-GM inequality
(ii) Cauchy-Schwarz inequality
(iii) Triangle inequality
(iv) The Fundamental Theorem of Arithmetic: Every integer n > 1 can be expressed as a product of primes in a unique way apart from the order of the prime factors.
(v) There exist infinitely many primes
(vi) Division algorithm: Let a be an integer and b a positive integer. Then there exists unique integers q and r, with 0 <= r < b, such that a = bq + r
(vii) If a and b are positive integers, then their greatest common divisor (gcd) is a linear combination of a and b, that is there exist integers s and t such that gcd(a,b) = sa + tb
(viii) If a and b are positive integers, and there exist integers s and t such that sa + tb = 1 then a and b are coprime.
Knowledge of the following theorems is not required and should not be used without proof: Euclidean algorithm,
Chinese remainder theorem, Wilson’s Theorem, Fermat’s little theorem, and Euler’s theorem.

The above is the syllabus for a subject called “Higher 3 Mathematics”. I would like you to set questions for a test paper that is about the content listed above. This paper must be extremely challenging. The paper should be out of 100 marks. It should contain 8 questions, with as many sub-questions as possible. Each question should be worth between 9 and 16 marks. All content mentioned as bullet points above must be used in the creation of the paper. As a rule of thumb for mark allocation, writing one mathematical step should grant you one mark.
The questions that you set should be more focused on applying to a particular context. This means that you should include some type of preamble for each question to set a context for the entire question. Do not set questions that make students merely regurgitate out answers that can be found in a textbook. For example, simple stating questions should not be set. Be as creative as possible when you are setting the questions.
For each question, you must define all polynomials, inequalities and mathematical identities in the preamble of the question.
None of the questions should be discussion questions. The questions should be mainly on proving and showing.