The archive
Computationally verified competition-level mathematics — TMUA, MAT, SMC, BMO1. All sheets are free PDFs; attempt under timed conditions before checking the methods.
Algebra
Polynomials, inequalities, and functional equations.
- 01 Quadratic in disguise: substitution and symmetric sums Methods:difference of squaresquadratic in disguisex + 1/x symmetry 33 questions Questions Answers
- 02 Paired products and denesting surds Methods:pairing factorsdenesting radicalsbiquadratic substitution 33 questions Questions Answers
- 03 Exponential substitution and the first inequalities Methods:exponential substitutionconstrained minimumsum-of-squares trick 33 questions Questions Answers
- 04 The t = x + 1/x device and AM–GM minima Methods:t-substitutionAM–GMreciprocal minima 33 questions Questions Answers
- 05 Symmetric systems and constrained inequalities Methods:x - 1/x formsymmetric systemsconstraint substitution 33 questions Questions Answers
- 06 Completing the square in two variables, and counterexamples Methods:surd substitutiontwo-variable completing the squaredisproof by counterexample 33 questions Questions Answers
- 07 Weighted AM–GM and harder proof structure Methods:weighted AM–GMCauchy–Schwarzequality cases 33 questions Questions Answers
Combinatorics
Counting, probability, and discrete structures.
- 01 Arrangements and the block method Methods:block methoddigit constraintsgrid counting 33 questions Questions Answers
- 02 Complementary counting and non-consecutive selection Methods:complementary countingat-least conditionsnon-consecutive selection 33 questions Questions Answers
- 03 Blocks with refusals, and circular arrangements Methods:internal block orderrefusal constraintscircular arrangements 33 questions Questions Answers
- 04 Binomial identities and double counting Methods:binomial coefficientsgreatest coefficientdouble counting 33 questions Questions Answers
- 05 Capped stars and bars, and inclusion–exclusion Methods:stars and barsinclusion–exclusionderangementscompositions 33 questions Questions Answers
- 06 Pigeonhole: guarantees, pairings and lattice points Methods:pigeonhole principlepairing argumentlattice parity 33 questions Questions Answers
- 07 Recurrences, tilings and invariants Methods:linear recurrencetiling recurrenceinvariant 33 questions Questions Answers
Number Theory
Divisibility, modular arithmetic, and primes.
- 01 Divisibility by rewriting, and reciprocal equations Methods:polynomial remainderdivisor pairsreciprocal Diophantine 33 questions Questions Answers
- 02 Factorising to force primality Methods:factorising for primalitydifference of squaresremainder trick 33 questions Questions Answers
- 03 Difference of squares and lcm–gcd pairs Methods:difference of squareslcm–gcd pairsminimal solutions 33 questions Questions Answers
- 04 Diophantine equations and the Sophie Germain identity Methods:Sophie Germain identityexponential Diophantineprime constraints 33 questions Questions Answers
- 05 Remainders, geometric sums and mod arguments Methods:polynomial remaindergeometric sum factoringmod 3 argument 33 questions Questions Answers
- 06 Simon's factoring trick, and gcd by Euclid Methods:Simon's favourite factoring trickEuclidean algorithmfactorial digits 33 questions Questions Answers
- 07 Counting divisors Methods:divisor-count formulaprime squaresparity of exponents 33 questions Questions Answers
Logic
Conditionals, proof techniques, and counterexamples.
- 01 Quantifiers: order and negation Methods:quantifier ordernegating quantifierscounterexample 33 questions Questions Answers
- 02 Necessary, sufficient, and equivalence chains Methods:necessary vs sufficientbiconditional chainsconverse 33 questions Questions Answers
- 03 Contrapositive and contradiction Methods:contrapositiveproof by contradictionsufficient conditions 33 questions Questions Answers
- 04 Counterexamples, and what actually refutes a claim Methods:counterexamplerefuting a universalcontradiction 33 questions Questions Answers
- 05 Spotting the flaw: the converse error Methods:converse erroraffirming the consequentflawed-proof analysis 33 questions Questions Answers
- 06 Induction, and where the base case fails Methods:inductionbase case failurestrong induction 33 questions Questions Answers
- 07 Equivalence, set claims and invariants Methods:logical equivalenceinvariant argumentextremal principle 33 questions Questions Answers
Sequences
Recurrences, series, and limiting behaviour.
- 01 Arithmetic sequences, and the sum as a quadratic in n Methods:nth-term formulaarithmetic series sumsum as a quadratic 33 questions Questions Answers
- 02 Binomial expansion, and Pascal's triangle as a sequence Methods:binomial expansionPascal's trianglegeneralised binomial series 33 questions Questions Answers
- 03 Geometric series, and when a sum to infinity exists Methods:geometric seriessum to infinityconvergence condition 33 questions Questions Answers
- 04 Recurrence relations, and solving them in closed form Methods:recurrence relationscharacteristic equationclosed-form solution 33 questions Questions Answers
- 05 Monotonicity, bounds and fixed points Methods:monotonicityboundednessfixed pointsfloor and ceiling sequences 33 questions Questions Answers
- 06 BMO1 capstone: constrained integer sequences Methods:extremal constructionconstrained integer sequencesalternating recurrences 33 questions Questions Answers
- 07 Mixed synthesis: every technique from Days 1–6 Methods:mixed synthesiscapstonetechnique selection 33 questions Questions Answers
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