Algebraic foundations
Simplify radicals and reason about rational and irrational numbers.
Loci, regions, and geometric methods
Choose between slicing, inverse membership, and vector or complex-number transformations.
- Find a region swept by a moving line: slices and inverse membership
- Root placement in an interval: discriminants, vertices, and endpoints
- Finding a locus: eliminate parameters and prove every point occurs
- Orthogonal projection vectors: line and plane formulas with applications
- Circle inversion in the complex plane: properties and worked problems
Integer problems and proof strategies
Use factorization, modular arithmetic, bounds, and rationality conditions.
- Three strategies for integer problems: factorization, modular arithmetic, and bounds
- Four approaches to Diophantine equations, with complete examples
- The rational root theorem: proof, candidates, and worked cubic equations
- Equating rational and irrational parts: proof and worked problems
- The largest integer that cannot be made from fives and eights
Limits and calculus
Understand limits, differentiation, integration, and equations with a parameter.
- Count real roots by isolating a parameter: a graphical method
- Three useful limit formulas, their meaning, and how to derive related limits
- The limits of sin x/x at zero and infinity, with a noncircular proof
- Integration by parts: formulas, proof, and four useful patterns
- The fundamental theorem of calculus in integral equations
- AM–GM: proofs, equality conditions, and five worked problems
Sequences, sums, and recurrence relations
Recognize telescoping, build recurrences, and organize proofs.
- Sums of squares, cubes, and fourth powers: formulas and proofs
- Telescoping sums: fractions, consecutive products, and arithmetic-geometric sums
- Ten recurrence patterns and how their solutions connect
- Probability recurrence relations: two University of Tokyo problems explained
- Mathematical induction: five proof patterns and worked examples
University of Tokyo past papers
Work through complete papers and compare approaches and difficulty.
- University of Tokyo science mathematics 2016: problems and solutions
- University of Tokyo science mathematics 2017: problems and solutions
- University of Tokyo science mathematics 2015: problems and solutions
- University of Tokyo science mathematics 2018: problems and solutions
- University of Tokyo science mathematics 2019: problems and solutions
- University of Tokyo science mathematics 2020: problems and solutions
- Five reasons I used Tetsuryokukai's University of Tokyo mathematics past papers
Common Test and Center Test
Study the structure and solutions of the Japanese entrance tests.
Study methods and books
Choose resources and build an effective practice routine.