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 difference of squaresquadratic in disguisex + 1/x symmetry 33 questions Questions Answers
- 02 Paired products and denesting surds pairing factorsdenesting radicalsbiquadratic substitution 33 questions Questions Answers
- 03 Exponential substitution and the first inequalities exponential substitutionconstrained minimumsum-of-squares trick 33 questions Questions Answers
- 04 The t = x + 1/x device and AM–GM minima t-substitutionAM–GMreciprocal minima 33 questions Questions Answers
- 05 Symmetric systems and constrained inequalities x - 1/x formsymmetric systemsconstraint substitution 33 questions Questions Answers
- 06 Completing the square in two variables, and counterexamples surd substitutiontwo-variable completing the squaredisproof by counterexample 33 questions Questions Answers
- 07 Weighted AM–GM and harder proof structure weighted AM–GMCauchy–Schwarzequality cases 33 questions Questions Answers
Combinatorics
Counting, probability, and discrete structures.
- 01 Arrangements and the block method block methoddigit constraintsgrid counting 33 questions Questions Answers
- 02 Complementary counting and non-consecutive selection complementary countingat-least conditionsnon-consecutive selection 33 questions Questions Answers
- 03 Blocks with refusals, and circular arrangements internal block orderrefusal constraintscircular arrangements 33 questions Questions Answers
- 04 Binomial identities and double counting binomial coefficientsgreatest coefficientdouble counting 33 questions Questions Answers
- 05 Capped stars and bars, and inclusion–exclusion stars and barsinclusion–exclusionderangementscompositions 33 questions Questions Answers
- 06 Pigeonhole: guarantees, pairings and lattice points pigeonhole principlepairing argumentlattice parity 33 questions Questions Answers
- 07 Recurrences, tilings and invariants linear recurrencetiling recurrenceinvariant 33 questions Questions Answers
Number Theory
Divisibility, modular arithmetic, and primes.
- 01 Divisibility by rewriting, and reciprocal equations polynomial remainderdivisor pairsreciprocal Diophantine 33 questions Questions Answers
- 02 Factorising to force primality factorising for primalitydifference of squaresremainder trick 33 questions Questions Answers
- 03 Difference of squares and lcm–gcd pairs difference of squareslcm–gcd pairsminimal solutions 33 questions Questions Answers
- 04 Diophantine equations and the Sophie Germain identity Sophie Germain identityexponential Diophantineprime constraints 33 questions Questions Answers
- 05 Remainders, geometric sums and mod arguments polynomial remaindergeometric sum factoringmod 3 argument 33 questions Questions Answers
- 06 Simon's factoring trick, and gcd by Euclid Simon's favourite factoring trickEuclidean algorithmfactorial digits 33 questions Questions Answers
- 07 Counting divisors divisor-count formulaprime squaresparity of exponents 33 questions Questions Answers
Logic
Conditionals, proof techniques, and counterexamples.
- 01 Quantifiers: order and negation quantifier ordernegating quantifierscounterexample 33 questions Questions Answers
- 02 Necessary, sufficient, and equivalence chains necessary vs sufficientbiconditional chainsconverse 33 questions Questions Answers
- 03 Contrapositive and contradiction contrapositiveproof by contradictionsufficient conditions 33 questions Questions Answers
- 04 Counterexamples, and what actually refutes a claim counterexamplerefuting a universalcontradiction 33 questions Questions Answers
- 05 Spotting the flaw: the converse error converse erroraffirming the consequentflawed-proof analysis 33 questions Questions Answers
- 06 Induction, and where the base case fails inductionbase case failurestrong induction 33 questions Questions Answers
- 07 Equivalence, set claims and invariants 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 nth-term formulaarithmetic series sumsum as a quadratic 33 questions Questions Answers
- 02 Binomial expansion, and Pascal's triangle as a sequence binomial expansionPascal's trianglegeneralised binomial series 33 questions Questions Answers
- 03 Geometric series, and when a sum to infinity exists geometric seriessum to infinityconvergence condition 33 questions Questions Answers
- 04 Recurrence relations, and solving them in closed form recurrence relationscharacteristic equationclosed-form solution 33 questions Questions Answers
- 05 Monotonicity, bounds and fixed points monotonicityboundednessfixed pointsfloor and ceiling sequences 33 questions Questions Answers
- 06 BMO1 capstone: constrained integer sequences extremal constructionconstrained integer sequencesalternating recurrences 33 questions Questions Answers
- 07 Mixed synthesis: every technique from Days 1–6 mixed synthesiscapstonetechnique selection 33 questions Questions Answers
No sheets match that search.