Equating rational and irrational parts: proof and worked problems
Learn when coefficients of an irrational number can be equated, prove that cube root of two cannot satisfy a rational quadratic, and avoid division by a possibly zero coefficient.
How to simplify nested square roots, with three worked examples
Recognize a sum and product inside a nested radical, keep the correct sign, and understand when rational radicands cannot remove the nesting.
Conditional probability: the formula, information protocols, and worked examples
Identify the conditioning event, distinguish independence from chronology, and solve family, sampling, noisy-report, bag, and tournament problems.
Integration by parts: formulas, proof, and four useful patterns
Derive integration by parts from the product rule and practice logarithmic, polynomial-trigonometric, polynomial-exponential, and cyclic integrals.
Triple-angle identities: remember, derive, and apply them
Check the signs and coefficients of sine and cosine triple angles, derive the tangent formula, and solve polynomial, complex-number, and integration examples.
University of Tokyo science mathematics 2020: problems and solutions
Complete solutions for cyclic inequalities, triangle-area regions, a rotating parametric region, subset-product sums, cone sweeps, and ellipse normals.
University of Tokyo science mathematics 2015: problems and solutions
Complete English explanations of six problems on swept regions, random strings, tangent curves, Fibonacci recurrences, binomial parity, and concentrating integrals.
The fundamental theorem of calculus in integral equations
Differentiate integrals with variable limits, retain initial conditions, and solve three examples including an integral recurrence.
Logarithmic differentiation: variable powers, products, and worked problems
Differentiate f(x) raised to g(x), handle absolute values and domain restrictions, and solve minimum, exponential-bound, and root-count problems.
Mathematical induction: five proof patterns and worked examples
Learn ordinary, two-step, and strong induction, descent, and forward–backward induction, then apply them to sequences, divisibility, inequalities, and integrals.
AM–GM: proofs, equality conditions, and five worked problems
Understand the arithmetic–geometric mean inequality, avoid a moving lower bound, and apply it to rational functions, geometry, and tetrahedron volumes.
Count real roots by isolating a parameter: a graphical method
Turn parameter-dependent equations into intersections of a fixed curve and a moving line. Worked examples cover quadratics, square roots, a semicircle, and the floor function.
University of Tokyo science mathematics 2016: problems and solutions
Six complete worked solutions covering exponential bounds, a tournament, projected areas, acute complex triangles, square-root digits, and a swept solid.
Reciprocal polynomial equations: symmetry, substitution, and applications
Use X=x+1/x to reduce palindromic equations, factor odd degrees, find the fifth roots of unity, and prove a symmetric quartic is positive.
Finding a locus: eliminate parameters and prove every point occurs
Use algebra and right-angle geometry to find three loci, carefully retaining parameter restrictions and excluded points.
How extra calculus and complex numbers can help humanities mathematics
Limits as checks, derivatives for optimization, and complex multiplication for rotations: four benefits and worked examples beyond a basic mathematics course.
Circle inversion in the complex plane: properties and worked problems
Understand w = 1/conjugate(z), prove how circles and lines transform, and solve three locus problems with complete domain checks.
University of Tokyo science mathematics 2019: problems and solutions
Complete solutions for an integral, constrained square geometry, an octahedron section, a nonsquare product, implicit-root limits, and a quartic-root locus.
Three strategies for integer problems: factorization, modular arithmetic, and bounds
Learn how to choose between three useful strategies, with complete solutions to integer equations, a parity proof, and a counting problem.
The largest integer that cannot be made from fives and eights
Solve the 5-and-8 coin problem by a five-column table, consecutive reachable values, and a proof of the two-denomination Frobenius formula.
Common tangents: five useful problem patterns
Work through parabola discriminants, matching tangent equations, quartic bitangents, circle distances, and counting tangents with a parameter.
Three useful limit formulas, their meaning, and how to derive related limits
Connect sine and exponential limits with local graph behavior, derive logarithmic and cosine limits, and keep the logical starting point of a proof clear.
University of Tokyo science mathematics 2017: problems and solutions
Six English solutions on trigonometric optimization, lattice walks, complex reciprocals, conjugate powers, common tangents, and a rotating equilateral triangle.
Three connected topics for University of Tokyo mathematics practice
Worked problems on probability recurrences, complex-plane geometry, and solid cross-sections illustrate three themes in a historical Tokyo exam-preparation guide.
University of Tokyo science mathematics 2018: problems and solutions
Six complete solutions on trigonometric calculus, binomial ratios, swept areas, cubic root placement, a complex parabola, and unions of rounded tubes.
Rock-paper-scissors probabilities: draws and the round of a final winner
Count equally likely hand patterns, find the draw probability for n players, and solve a three-player elimination game by paths and recurrence relations.
Perfect-score practice: three lessons from the former Center Test
A historical study account explains review-time buffers, choosing reliably checkable questions, and three calculation shortcuts for areas, sums, and tetrahedron volumes.
Areas enclosed by parametric curves: cardioids, astroids, and a check
Set up signed integrals with the correct branches, calculate two complete examples, and verify them using the parametric area formula.
Four approaches to Diophantine equations, with complete examples
Solve integer equations using a particular solution, the Euclidean algorithm, factor pairs, discriminant bounds, and ordered positive variables.
Factoring 21 with Shor's algorithm: qubits, periods, and retries
Follow a normalized order-finding example, understand Fourier interference and its limits, and see why base five fails before base two yields the factors three and seven.
The rational root theorem: proof, candidates, and worked cubic equations
Find every possible rational root of an integer-coefficient polynomial, prove the divisibility conditions, and use them to solve or rule out roots of cubics.
Five reasons I used Tetsuryokukai's University of Tokyo mathematics past papers
A historical review of alternative solutions, strategy explanations, model answers, yearly paper sets, and detailed difficulty ratings.
Telescoping sums: fractions, consecutive products, and arithmetic-geometric sums
Find a discrete antiderivative so that middle terms cancel. Three complete examples show how to choose the right difference and check its endpoints.
Root placement in an interval: discriminants, vertices, and endpoints
Find parameter conditions for two roots, exactly one root, or at least one root in an interval, with four complete quadratic and cubic examples.
Find a region swept by a moving line: slices and inverse membership
Solve the same locus problem in two ways: find each vertical slice, or test whether a point has an allowed parameter. Includes boundary checks, animations, and a two-parameter example.
Which primes have the form a squared + ab + b squared?
Prove the prime representation criterion using Eisenstein integers, polynomial quotients over finite fields, Euclidean division, and norms.
The limits of sin x/x at zero and infinity, with a noncircular proof
Use the squeeze theorem for both limits, distinguish radians from degrees, and justify the sector-area argument without assuming trigonometric derivatives.
Sums of squares, cubes, and fourth powers: formulas and proofs
Derive power-sum formulas from telescoping differences, verify the cube formula by induction, and sum consecutive products without expanding them.
Orthogonal projection vectors: line and plane formulas with applications
Derive orthogonal projections using dot products, find a perpendicular foot, and reflect a point across a plane without needing its full geometry.
Probability recurrence relations: two University of Tokyo problems explained
Use transition diagrams, symmetry, and parity to solve a rolling tetrahedron problem and a random walk through nine triangular rooms.
Ten recurrence patterns and how their solutions connect
Solve affine, reciprocal, logarithmic, exponential, second-order, polynomial, variable-coefficient, coupled, partial-sum, and implicit recurrences.