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.
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.
The fundamental theorem of calculus in integral equations
Differentiate integrals with variable limits, retain initial conditions, and solve three examples including an integral recurrence.
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.
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.
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.
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.
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.
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.
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.
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.